Benjamin Ho Economics of Apologies

8/6/2019 Benjamin Ho Economics of Apologies

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The Economics of Apologies:Theory and Experiment

Benjamin Ho*

Stanford Graduate School of Business

Revised: April 2006First Draft: May 2003

ABSTRACT

Apologies are a previously understudied social institution integral in the maintenanceof social relationships. Their application ranges from corporate culture to political systems tolegal settings. This paper formulates a game theoretic signaling model using rational agentswith two-dimensional type that is general enough to encompass a broad class of gameswhere apologies can be used. I characterize the equilibrium and establish an existence resultthat extends single-crossing. The theory is then tested using a novel variant of the trust gameexperiment confirming all predictions. I use these findings to assess the use of apologies inmedical malpractice litigation.

Keywords: apologies, remorse, signaling, trust game, empathy, attribution theory

JEL Classification: C72, D82, L14, D23

* Author can be contacted by e-mail at [email protected] by phone at 650-867-8270, or bymail at GSB-PhD 518 Memorial Way Stanford, CA 94305. Thanks to Edward Lazear,Douglas Bernheim, John Roberts, for exemplary advising, Chip Heath, Andrzej Skrzypacz,Emeric Henry, Florian Ederer, Luz-Marina Arias for extensive comments, and toparticipants of the Theory seminars at Stanford Economics and the Stanford GSB. Allremaining mistakes are mine. My apologies.


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1 Introduction

Paul had a problem. Several weeks ago, he made a date to play tennis withhis friend Amy at noon on Saturday. Paul arrived at the courts on time, butAmy shows up one hour late. Paul is angry but after Amy apologiesprofusely, Paul readily forgives her and they both enjoy the tennis match.They make a date for the following week, whereupon Amy is again late,again apologizes, and again is forgiven. On the third week, Amy is lateonce more, and at this point Paul is fed up. Talk is cheap, why doapologies have any meaning?1

A common opening for papers about apologies across the social sciences is to

note the dearth of articles about apologies in their respective field. I begin no

differently. A search of Econ-lit produces no results on the subject of apologies, yet

this social institution has wide ranging significance. Beyond the use of apologies in

daily interpersonal interactions, apologies appear in organizational design, political

reputations, legal litigation, international relations, corporate governance, and

beyond. In an interconnected world where economic actors are all embedded in a

network of relationships (Granovetter 1985; Sen 1977; etc.), apologies act to restore

frayed connections.

The fundamental goal of this project is to provide a rational framework for

understanding a social institution that is presumed to be based on emotion. Why do

people apologize? When do people apologize? What is the cost of an apology? What

is the benefit? Why does the benefit deteriorate with time? If apologies mend

relationships and talk is cheap, why is it that people do not always apologize?

I focus on interactions where exogenous factors make payments infeasible,

either due to legal reasonse.g. the inability for a politician to bribe the

electorateor from social normsgifts2 between friends tend to be limited to

symbolic gestures. Also, I consider one-sided interactions, where payoffs for both

principal and agent are dependent on hidden information about the agents type.

I avoid psychological assumptions to allow for applicability to potentially

highly rational actors like politicians and governments. If shame and guilt are the

true motivation of apologies, this paper can address why these emotions arose, and

why we pass on notions of remorse to our children. This paper also defines

1 The story is true, but the names and details have been changed to protect the innocent.2 There is a gift giving literature where agents choose inefficient gifts in order to signal type(Camerer, 1988; Prendergast and Stole, 2001, etc.). My approach here differs by beingspecific to the apology context and by being more broadly applicable to situations wheregifts are disallowed and for a much broader class of interactions.


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behavioral notions such as mistake, sympathy, empathy, situation, disposition, and

intention formally, making them tractable for game theory.

The model I propose is general enough to encompass a broad class of principal-

agent interactions and a broad class of different types of apology. In all such games,

the model gives universal predictions, e.g. apologies matter more in longerrelationships and when there is more uncertainty, also types that yield higher utility

for the principal always apologize more. With the model in hand, important policy

questions can be addressed. For example, the emergence of Im sorry laws in

Texas, California, Massachusetts, Florida and other states make apologies

inadmissible in courts. Apologies are also a prominent issue for politicians; the

popular press constantly asks why politicians never apologize. By formalizing

terminology and systems that have previously only been informally discussed, I

seek to clarify these issues.

An apology is a costly signal which restores relationships not just because it is

cheaper for good types to apologize (as in Spence, 1974, etc.) but also because good

types fail in different situations than bad types. Apologies have signaling value even

if single-crossing is not met. I then show that this framework serves as a reduced

form for a broad class of cheap-talk apologies, and give three example

mechanismscontracting, status, and empathythat endogenize the cost. Finally, I

consider two applications: a novel trust experiment that validates the theory, and

apologys legal ramifications.

1.1 Model ConceptsEveryone has some notion of what it means to apologize. To crystallize ideas, I

lay out the framework that I use for the remainder of the paper. An apology occurs

between two actors, a principal and an agent. One can think of the players as part of

a larger community where the principal needs a task accomplished and solicits the

agent. Agents are ordered by how well suited they are to the principals needs. In a

political game, this would be closeness of ideal point. In a standard principal-agent

framework, this would be the agents productive ability. In a simple divide the

dollar game, this would be a Fehr-Schmidt (1999) fairness or Becker (1976) stylealtruism parameter.3 Social psychologists would call this type, the agents

disposition, and I will speak of good dispositions and bad dispositions.

3 One might call this parameter,sympathy. The American Heritage Dictionary (2000) definessympathy as A relationship or an affinity between people or things in which whateveraffects one correspondingly affects the other. In Beckers model, a high altruism parametermeans that high utility for the other yields high utility for ones self.


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Interpretation of type differs by context. The commonality is that higher types yield

better outcomes, i.e. higher utility, for the principal.

In the example above, Paul is the principal and needs an agent to meet for

tennis. Paul partners with the agent, Amy, based on his beliefs regarding her

punctuality, i.e. her type. During the first two apologies, Paul is willing to attributethe lateness to random justifiable events, e.g. traffic. After the third time, Paul

concludes that he was wrong with his earlier attributions and ends the relationship.

The broad sketch of the game proceeds as follows: nature draws the agents

type ),( consisting of two dimensions: is intrinsic to the agent and stable

across time, while represents thesituation and is uncorrelated across time. There

is some production technology that maps the agents type into an outcome, ),( y ,

for the principal. The information structure is such that the agent knows her type,

and hence also the outcome, but the principal observes only the outcome. The agent

can then put forward an apology that updates the principals beliefs about the

agents disposition. Finally, the agent receives continuation payoff, ),( bv as a

function of the principals posterior beliefs and the agents disposition itself.

(Insert Figure 1a about here)

The model is a signaling game with an embedded moral hazard problem. The

action of the apology mechanism occurs with the signaling, and thus I consider a

reduced form of the game that encapsulates the details of the production technology.

However the production technology is important because the agent must be given

the opportunity to make a mistake. I consider an apology to be any act of

communication expressing regret or asking pardon for a fault or offense

(American Heritage Dictionary, 2000). A mistake for homo economicus is the

difference between what is ex post optimal and what is ex ante optimal. Mistakes

are modeled using a random move by nature, because alternatives such as off

equilibrium play or trembles presumes that mistakes occur with near zero

probability, whereas in life, apologies occur quite often.4

The production phase itself will be explored in some detail in Section 2.2, but

for now, it suffices to say that the agents type ),( is transformed into an

outcome that yields utility ),( y for the principal, where ),( y is increasing in

4 There is an alternative use of the term Im sorry in the English language where no fault isacknowledged, e.g. I am sorry your grandmother died. By the above definition, this is notstrictly speaking an apology, as the agent is not taking responsibility for the bad outcome.However, the ideas have somewhat become conflated, and thus I explore these partialapologies in the appendix.


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. After the production phase, the principal eitherattributes a bad outcome to the

agents disposition, a low , or to the agents situation, a bad draw of. How this

dilemma resolves is the subject of attribution theory (Ross, 1977; etc.). An apology

is the agents attempt to shift the principals attribution from disposition to situation.

A key component of an apology is its value in the restoration of relationships(Tavuchis, 1991; Ohbuchi et al., 1989; McCullough et al., 1998; etc.). The larger

ecology of apology is beyond this papers scope, but since I assume that actors want

to maintain relationships with those of higher type, the principals beliefs regarding

the agents type is a measure of the strength of the relationship.

Finally, as this paper considers situations where payments between actors are

disallowed, the payoff is in the form of a continuation value based on the probability

the principal decides to continue the relationship, i.e. forgive. More specifically, it is

based on the probability that the principal stays with the same agent for this task,

rather than switch to a different one. Since I focus primarily on the apology,

forgiveness is left in reduced form.

1.2 ApplicationsTo emphasize the import of apologies in many economic and political

interactions, I highlight six applications in particular, and will return to them in the

discussion after developing the model:

Interpersonal Relationships: Apologies are a common occurrence in everyday life,

particularly in the maintenance of friendships. Empathy is particularly important in

this context. The model also addresses cultural differences in apologiesAsians

apologize more than Americans (Takaku et al., 2001; etc.)or gender differences

women apologize more than men (Gallup, 1989 in Tavuchis, 1991).

What psychology literature exists focuses on experimentally validating stylized

facts. An apology by the agent reduces the anger the principal feels toward the agent

as well as the principals desire to punish (Ohbuchi et al., 1989). In tasks where the

agent is less responsible or where the offense is less severe, the apology is rejected

less often (Bennett and Earwalker, 2001). Apologies are almost always accepted

(Mullet et al., 1998; Bennett and Dewberry, 1994). Forgiveness occurs more oftenin closer relationships (McCullough et al, 1998).

The results of all the psychological experiments I found in the literature were

consistent with the findings in the model presented below.

Organizations: The prevalence of apologies in various organizational settings is

indicative of differences in task assignment, risk taking, turnover, conflict


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resolution, complementarities in production, etc. A model of apologies offers insight

into cultural differences in organizational design. Lee and Tiedens (2001a) find that

within an organization, when individuals of high statusthose with controlmake

excuses for their behavior, they suffer a decrease in status.

Corporate Governance: CEOs are expected to be responsible to shareholders.When performance is low or scandal arises, should an apology be expected? Does

an apology carry any weight? Lee and Tiedens (2004) find that certain kinds of

attributions for past performance found in company annual reportseffectively

kinds of apologycan predict a firms stock prices one year out.

Politics: There is a stylized fact that politicians never apologize. Consider, Bush on

Iraq, or Clinton on Lewinsky. Tiedens (2001) conduct an experiment where she

constructs two videos by splicing interview footage, one where Clinton appears to

apologize regarding the Lewinsky affair, and one where he appears angry. Subjects

who saw apologetic Clinton liked him more, while subjects who saw the angry

Clinton liked him less and complained about how Clinton never apologized.

However, on questions of leadership, ability, and importantly, whether you would

re-elect, the angry Clinton fared better. Their result is robust to choice of politician

and the crime. Section 3 explores these apologies in greater depth.

Governments: An apology by a government is important either between the

government and its people (e.g. South African apartheid, Japanese internment, or the

United States civil war), or between governments in international relations (the

difference between the German and Japanese response to World War II).

The South African Truth and Reconciliation Commission effectively lowered

the cost of apology so that apologies can be made and relationships restored. Simple

punishments would have left relationships broken.

Law: In recent decades, apologies have become increasingly important in litigation

damages. Many US states have recently passed laws that bar the use of apologies as

evidence. Apologies are especially relevant in medical malpractice, as a vicious

circle has arisen. Doctors are afraid to apologize because of the risk of lawsuits.

Patients are more likely to sue due to anger for not receiving an apology (Taft, 2000;

Cohen, 2000; Latif, 2001; Cohen, 2002). See the Discussion for more details.

1.3 MechanismsThe psychology literature provides many properties of apologies, but is largely

deficient at describing the process. A typical reason given for why people apologize

is limited to negative affect alleviation. People feel guilty, and an apology

removes guilt. This view dates to a Freudian model of behavior where humans have


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stocks of emotione.g. guiltwhich when accumulated, causes distress until

emptiede.g. by apologizing (Cialdini et al., 1975; Cialdini and Ascani, 1976; etc.).

Tavuchis (1991) has a sociological treatment of apologies in which he sees

apologies as a social system designed to maintain relationships and establish

membership in community. Tavuchis describes an apology as a social exchange, adevice that paradoxically restores social order without amending the transgression.

Nothing material has been exchanged, yet the relationship has changed. The pain of

an apology is created by the social system of shame that accompanies it.

These psychological and sociological models of behavior while not consistent

with economic rationality, may be accurate. However, the model proposed is more

teleological. In the mode of Frank (1989), culture provides devices such as guilt,

remorse and shame, in order to help people play the rationally optimal strategy, in

this case, to maintain relationships. The rational actor assumption is retained for

modeling purposes using Friedmans (1953) as-if justification.

Attribution theory from social psychology, though incomplete, is more useful

for illuminating apologies. Attribution theory concerns itself with settings where an

outcome is observed, e.g. Amy is late for a tennis match, that could have two

possible causes: 1) dispositional, Amy is lazy and inconsiderate, or 2) situational,

Amy was held up by unexpected traffic. Psychological theories about the

fundamental attribution error predict that individuals attribute too much blame to the

disposition of the actor and not enough to the situation (Heider, 1958; Ross, 1977;

Jones and Nisbett 1972).5 Weiner et al. (1987) applies attribution theory to the

process of apologies, and in experiments, finds that apologies that attribute bad

outcomes to external uncontrollable situations, , are more likely to increase liking

than apologies that attribute bad outcomes to internal controllable dispositions, .

In the models presented, an effective apology shifts the principals attribution of the

cause of the bad outcome from the agents disposition to the external situation.

Finally, note that the economic model presented here focuses on the apology

mechanism itself, a self-contained event that does not depend on punishment or

penitence. It does not use repeated game punishment scenarios6 or reputational

cheap talk. 7 Reputation is based on type rather than equilibrium, because apologies

occur often, and thus it is better for mistakes to occur on the equilibrium path.

5 Though not addressed in this paper, principals in this paper rationally commit thefundamental attribution error, in order to give agents incentive to provide costly information.6 Axelrod (1984), Green-Porter (1984), Abreu, Pearce and Stachetti (1990), etc.7 Ottaviani-Sorensen, 2004; Jackson-Sonnenschein, 2003;Hauser-Hopenhayn, 2005


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2 Base Model

2.1 SetupThe apology game has two players, a principal and an agent. I begin analysis

with two periods, but consider dynamics with more periods in section 2.5. Thetiming within each period is as follows and summarized in Figure 1. First, nature

sets the agents disposition, },{ BG 8 and let the prior ]Pr[ Gp == . Nature

also sets the agents situation as , with probability measure )(F and is

a finite set. The agents disposition is the component of type that is correlated across

periods. The agents situation is the component of type that is uncorrelated across

periods. The agent knows both dimensions of her type, the principal knows neither.

Then, through some production technology, the agent produces an output that

yields payoff ),( y for the principal, as a function of her type, ),( . Of coursethis reduced form specification of production leaves out potential moral hazard.

Examples of production technologies that incorporate an action by the agent yet

yield such a reduced form are given in the next section.

Both players observe the realized outcome, at which point, the agent can

choose whether to apologize or not, which in the base model is a costly signal. The

principal updates his posterior beliefs about the agents type, ),,( pyab , using the

realized outcome and the apology.

(1) ),|Pr(),,( yapyabG

==

Finally, the principal chooses either to stay with the same agent or to take his

outside option. If the principal stays with the same agent, the period is repeated with

the same but a new . If the principal chooses the outside option, he gets a new

agent with a new draw of . At the end of the first period, the principal learns the

prior outp that the outside option is a good type. Since the principal maximizes his

payoffs by interacting with higher typed agents, the principal chooses the agent over

the outside option if the probability she is a good type is higher than the probability

the outside option is a good type:

(2) ]),,(Pr[)),,(( outppyabpyab >=

8 All the results hold for a continuous distribution of types, but the notation is ratherunwieldy, since f.o.s.d. does not provide a complete ordering.


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The utility of the agent is linear in three components: the agents utility from

production, the cost of apologizing, and the agents discounted payoff from future

interactions with the principal:

(3) )),,,((),,(),(),|( pyabvacuaUA +=

To make sure the agent wants to stay in the game, I normalize 0),( >u .

Also, let }1,0{),( a where a one indicates an apology. The model does not

distinguish between a non-apology, and inaction. Inaction is interpreted as a

message of 0=a .9. I normalize the cost function such that 0),,0( =c and let

),,0(),,1( cc > .

The principal cares only about maximizing his output for each period

(4) +==t

tttP yyyyU ),(),(),()( 2211

Type is defined in such as way such that ),( y has the property that

),( yE is increasing in . Note that all I require is a production technology

where agents can be ordered by how well they produce for the principal.

If the agent stays in the relationship, she gets payoff ),( 2u in the second

period, if she is terminated she gets payoff 0 . Thus the first period continuation

payoff in this simple two period model would be:

(5) ),()),,(()),,(( 211 upyabyabv =

We will see that all that is necessary for the results to hold is ),( bv increasing

in b , a condition which is satisfied here.10 I consider a pure strategy Markov perfect

equilibrium where the state variable is the principals beliefs about the agents type.

Existence of such equilibria is guaranteed in finite games by Maskin and Tirole

(1990). Before continuing to the analysis, I first provide several sample production

technologies in section 2.2 and several sample cost function in section 2.3.

9 One might argue that a richer message space may offer different results, but especially forcheap talk models, as Crawford-Sobel (1982) shows, sender-receiver games with imperfectlyaligned preferences, allow only finite number of different messages, and there are manyequilibria. Effectively, I consider the equilibrium where there are two messages.10 It should be said that we do apologize even in situations where we never expect toencounter the counterparty again. One could argue that we do this for the same reason wetip, that we behave, as if we will meet again, or that we follow some behavioral heuristicor script (Kahneman and Tversky, 1974; etc.). v(b,), however, is completely general andcan just as easily accommodate psychic reasons for wanting to be liked.


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2.2 Examples of Production TechnologiesSince apologies occur in a wide variety of social situations, the model is

designed to accommodate a broad class of games. I embed these games into the

production phase of a broader apology game that adds an apology action and

repeated play. This section gives several examples of games that the productionphase could represent.

Figures 1b, 1c and 1d present three game structures for the production

technology.

(Insert Figure 1b about here)

In the first, the agent must choose the action before she learns the realization

of. In this view, represents the agents intentions, before the state of the world,

, is realized. The state of the world might represent new information about the

situation or new thinking on the part of the agent. The optimal x given would be

different than the ex ante optimal . This difference represents regret. An apology

here signals that the outcome resulted from a bad situation rather than bad

intentions.

(Insert Figure 1c about here)

In the second view, x is chosen after both and are realized. However,

only is persistent, and thus for future interactions, the principal only cares about

. In this view, represents a temporary mood as in Bernheim and Rangel (2004).

The principal only cares about the agents true disposition, , but cannot easily tell

given the confound of. An apology indicates that despite the bad outcome in the

past, the principal can expect a good disposition in the future.

(Insert Figure 1d about here)

In the third view, represents the old type, while represents new thinking.

However, the hidden action, x was chosen before the new thinking was realized.

Again, the principal only cares about future interactions, thus an apology signals the

difference between the old and the new types.

The necessary common property for these production games is that the agents

type ),( is translated into an outcome that yields utility ),( u for the agent

and utility ),( y for the principal, where ),( y is increasing in via some

action, x , by the agent. In most examples of interest, outcome is a function both of

type and the agents action: ),,(~ xy . However, in equilibrium the agents action


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is uniquely determined by her type, ),(* x , once I ensure the outcome function

),( y has the necessary properties, I will restrict focus to the reduced form:

(6) ),),,((~),( * xyy =

Typically, the natural specification of agents utility is as a function of action,

output, and type: ).,,,( yxu So long as ),,( xy is invertible in x , it is a

simple matter to rewrite ),,,( yxu as ),(~ yu :

(7) ),,),,,((),(~ yyxuEyu =

Effectively, instead of choosing an action, the agent is choosing an expected

output level for the principal. The necessary assumption, then, is a simple restriction

on the utility function, that ),(~ yu has increasing differences in y and . Either

higher agent types value the principals utility more, or it is easier for higher agent

types to provide principals with higher utility. Thus, taken in isolation, by Topkis

Theorem, an higher agent maximizing such utility yields highery .

Recall, however, that the agents choices are being embedded in the larger

apology game. Thus, in the larger game, it is necessary for the agents full utility,

(8) [ ])),,,((),,(),(~)|( pyabvacyuEyUA +=

to have increasing differences in y and . Since the continuation value is

composed of the production utility and the cost, a simple sufficient condition is thatthe production utility is increasing in and that the cost function is independent of

. Alternatively, one could assume that the agent has a sufficiently low discount

rate for the future such that the supermodularity of the present is preserved.

This specification for the utility function and production technology is

awkward, but it captures many common moral hazard problems. Some examples

may serve to clarify.

The first example is the standard moral hazard with high and low productivity

where x represents effort. The agents cost of effort is increasing in x , but higher

types have a lower marginal cost of effort.

(9)

+=

=

xxy

xyxu

),(

),,(2

Assuming the noise term, has mean zero, this expression yields


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(10)

)(),(~

2 Varyyu

+=

which can be differentiated to show that it satisfies increasing differences.

A second example is a political game, with a uni-dimensional policy space, and

represents the agents choice of policy. The principal has an ideal point of zero,

while the agents have ideal points away from zero, and /1 represents the agents

ideal point. This functional form can also model the Paul and Amy interaction,

where x is the choice of departure time, and is the amount of traffic:

(11)2

2

)(),(

)1

(),,(

+=

+=

xxy

xyxu

A third example might have as an altruism parameter, and the task is some

noisy gift giving game, where the agents choice is how much the agent gives to

the principal, but the agents choice is obscured by noise .11

(12)

+=

=

xxy

xyyxu

),(

),,(

The point again of ensuring supermodularity is to guarantee that higher types,

, make choices that lead in expectation to higher utility for the principal, y .

Having established this requirement, I return to the reduced form specification.

2.3 Examples of Cost FunctionsThe analysis of this paper is designed to hold for fairly general production

technologies and selection functions. So long as moral hazard is properly

maintained and the existence conditions are satisfied, a wide array of cost functions

is allowed.12

Relationships can be restored using a wide variety of different apology

mechanisms. When people use the words I am sorry, they mean different things in

different circumstances. An apology can represent a tangible cost, such as in

emotional pain,13 time spent or an acknowledgment of responsibility with third party

11 This formulation would yield a corner solution but that does not affect the conclusion.12 The cost paid for an apology is given exogenously instead of being chosen by the agent.Allowing a to be a continuous variable would not substantially change results.13 The costs can be entirely behavioral, such as through norms of shame that surround thevery act of giving an apology, mitigated by alleviation of guilt. Tavuchis (1991) argues thatsociety makes it painful to apologize through norms of social disapprobation, because thepain is what gives apologies meaning. The model here is consistent with this view of


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legal ramifications. An apology could entail the giving of an account or excuse

where a false apology must face sincerity detection14 An apology can represent a

commitment to work harder in the future. Or an apology can represent a loss of

status as is often the case with politicians.

Proposition 3 in Section 3 shows that so long as the assumptions of the modelhold, all of these mechanisms must share the same reduced form in order for

signaling to be achieved. and thus the propositions in this paper holds for such

divers apology types. However, a more structural view of each of these underlying

mechanisms can yield more precise insight, a full discussion of which is left for the

appendix.

2.4 Analysis of Base CaseBefore proceeding, it is important to note where the signaling is coming from.

Unlike standard signaling models, the following results all hold equally well if both

cost and payoffs were independent of . Standard signaling models are based on

setups where it is cheaper for good types to apologize or where good types get

higher payoffs because they value the relationship more. While this could be true

here, the innovation is that apologies can also work because a bad outcome for the

principal occurs in different situations for good agents than for bad agents. This is

articulated precisely in the existence results in Proposition 2. I first assume existence

and consider some properties:

Proposition 1:In the apology game with c(a,,) increasing in a, v(b,) increasingand continuous a.e. in b, and Ey increasing in , any pure strategy Markov perfect

equilibrium using beliefs about as the state variable where both apologies and

non-apologies are equilibrium outcomes at a given y has the following properties

a. Principal beliefs b are strictly increasing in apologies, a , outcome, y,prior, p and agents type

b. Probability that an agent apologizes, ]|1Pr[ ya = , is strictly increasing inagents type . The probability of apologies is low for extreme values ofp

andy 15 and maximized for intermediate values.

apologies. Again, although there is considerable psychological and sociological evidencethat such mechanisms of shame and guilt are at work, I focus on economic explanations.14 See Frank (2004) for an excellent account on sincerity detection in human society.15 The relationship between a andy depends on an additional regularity condition.


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c. The outcome for the principal, y, is strictly increasing in the agents type, . It is low for extreme values ofp and maximized at intermediate ones.16

This relationship is summarized in the following correlation matrix, Table 1:

(Insert Table 1 here)

Note that the Proposition does not place any special restrictions on the cost

function or the payoff function. However, it does depend on existence. The

conditions for existence are given in Proposition 2 below.

Proof part (a): I examine the relationships from Proposition 1, one at a time.

Working backward, an agent apologizes if and only if she gets higher payoffs from

apologizing:

(13) 0)),,,0((),()),,,1(( pybvcpybv

or rearrange to get the apology condition:

(14) ),()),,,0(()),,,1(( cpybvpybv .

The first thing that can be noted is that apologies always help strengthen/repair

the relationship. Recall that ),,( pyab indicates the strength of the relationship.

Since ),( bv is increasing in b , and 0),( >c , an apology will only be tendered

if ),,0(),,1( pybpyb > . Returning to the example, one can think of Paul being

upset with Amy when she is late (a bad draw of y ). Until an apology is tendered,

Paul is unhappy with Amy and has belief ),,0( pyb . After an apology, the

relationship is partially restored up to ),,1( pyb .

Now, let us turn to the principals beliefs. First note than in equation 14, the

l.h.s. is independent of and so for any given , there is a )(* c where the

equation is satisfied with equality. Then define:

(15)

)}(),(:{

)}(),(:{

*

*

BBA

B

GGA

G

cc

cc


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(16)}~),(:{

}~),(:{~

~

yy

yy

By

B

Gy

G

=

=

The posterior, ),,( pyab , is derived from the prior using Bayes rule:

(17)

)1)(()(

)(

],0|Pr[),,0(

)1)(()(

)(

],1|Pr[),,1(

00

0

11

1

pFpF

pF

ypyb

pFpF

pF

ypyb

y

BB

y

GG

y

GG

G

y

BB

y

GG

y

GG

G

+

=

==

+

=

==

From Equation 17, a few simple relationships can be derived.

(18) 0/ > pb

In words, the stronger the relationship before the interaction, the stronger the

relationship will be after the interaction, ceteris paribus.

Similarly, by the definition of , is increasing in . Thus using a chain rule

argument

(19) 0/ > yb

The better the outcome, the more the principal likes the agent.

Finally, since higher means both higher y and as I will establish, higher]|1Pr[ ya = , higher means higherb . Better types are in expectation better liked.

Proof part (b): The next result is that good types apologize more than bad types.

Rearranging terms from Equation 17, ),,0(),,1( pybpyb > if and only if

(20)

],|1Pr[

],|1Pr[1

],|1Pr[

],|1Pr[1

)(

)(

)(

)(0

0

1

1

ya

ya

ya

ya

F

F

F

F

B

B

G

G

y

GG

y

BB

y

GG

y

BB

=

= for good types and pb < for bad types, the principals beliefs become more

accurate with time.

To see how these results can be utilized let us return to the tennis example. Paul

is choosing between two potential tennis partners Amy and Alice, and wants to play

with a partner of a good type. Pauls prior is that Amy is more likely to be a good

type than Alice, so he partners with her. After the first instance of Amys tardiness,

Paul has a dim view of Amys type, but her apology restores his beliefs closer to but

still below his initial prior. High enough, though, so that Amy is still more likely to

be better than Pauls outside option, Alice. However, a second failure and apology

lowers Pauls confidence in Amy further. The third failure lowers Pauls belief in

Amys type so that even after an apology, Pauls belief that Amy is good is now

lower than Pauls prior that Alice is good. Paul ends the relationship with Amy.

Successes strengthen relationships. In case of failure, apologies restore therelationship but only imperfectly so. An apology after one failure may restore the

relationship sufficiently to continue the game, but a succession of failures can leave

b low enough for the principal to end it.

3 Cheap ApologiesThe model presented in the previous section is effective at explaining many of

the characteristics observed in apology interactions. However, its reliance on an

externally given cost function leaves the modeling unsatisfying. In this section, I

18 The agent is maximizing the sum of production utility, cost function and continuationvalue. The production utility and the cost function is independent of prior. The continuationvalue is increasing in posterior beliefs and thus in prior as well. Therefore, from Proposition1, for high enough p , apologies are decreasing in p , so the cost of apologizing is

decreasing in p , so in sum, agent welfare is increasing in p .19 Assuming that the agent was initially selected as the principals acquaintance with thehighestp and that the outside option is the principals acquaintance with the second highest

p, this assumption seems reasonable for a large enough population of potential partners.


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return to the question originally posed: if talk is cheap, why do apologies have any

meaning? I now assume that an apology is simply a message }1,0{a , and its

meaning determined in equilibrium. The models in this section all fit into the basic

apologies framework from Section 2, but each serves to unpack the black boxes that

the cost, )(c , represents. It is useful to note that these mechanisms are necessary.

Proposition 3: If the cost of apologies in the base model is set to zero, there is no

Markov perfect equilibrium in which apologies reveal information.

This proposition follows because if apologies were used differentially by the

two types, one message would shift beliefs higher. Then both types would only send

that message, since given the refinement, only beliefs carry to the next period.

I propose instead several modifications that endogenize the cost in the base

model. Though each variation can be modeled in reduced form as a cost function, it

is instructive to consider how introducing different game mechanics can give

meaning to cheap apologies.

Third party enforcement: The simplest mechanism creates a tangible cost by

introducing a third party. Assume a legal system that can impose a fine or legal

sanction on the agent. The probability that the fine is imposed depends on the

agents guilt which cannot be known with certainty without knowing . However,

whether an agent apologizes can be used as evidence. The system thus imposes a

cost as a function of the apology, the outcome and the state of the world.

Lying: The word apology derives from the Greek for story. An apology is an

example of account giving or excuse giving. The cost of account giving is based on

the probability the lie will be found out. Amy claims she was late due to traffic, but

there is some chance Paul will hear a traffic report and catch her lie. Or

alternatively, many argue that humans are evolved to be poor liars and dishonesty

can be detected by facial cues (Ekman, 1969; Frank, 1989). In either case, the

possibility exists that a lie will be found out, and the cost represents the punishment

associated with lying, determined socially outside of this model.

Formally, one could endogenize the cost of lying or excuse giving by having an

apology be a message that reports the state of the world that is initially

unobserved by the principal. However, with some probability, false reports are

caught which leads the relationship to be severed. Honest reports are never punished

and thus costless. A good agent would report truthfully as it is costless to do so.

If a bad agent reported truthfully, the principal knowing ),( y and could

back out the agents type, thus if the bad agent wants to be perceived as a good


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agent, she must lie. The cost of lying depends on the probability of getting caught

which depends on and also the incentive to lie which depends on . A good

agent would always be able to tell the truth and incur zero cost for such an apology.

Commitment: This possibility could be termed the Im sorry I will never do it

again apology. It depends on the principals ability to commit to firing an agent heknows to be good with some positive probability. The resulting fool me once,

shame on you, fool me twice, shame on me equilibrium uses the apology message

as a promise to never do it again. The principal maintains incentive compatibility by

punishing a failure that follows an apology more severely than if there was no

apology at all.

Status: A common reason given that it is difficult to apologize is that an apology

entails a loss of status, a shift of the principals beliefs in the status dimension. One

could argue that the reason women apologize more than men arises from

evolutionary pressure that made status more important for men as men need status to

compete for mates. Since status matters relatively less for women, they can

apologize more. The same can be said for Asian cultures relative to Western ones. If

Asian cultures value group preference alignment more than individual ability, then

apologies will be more prevalent in Asian cultures.

One would expect status concerns to be especially critical to political

apologies. Tiedens (2001) experimentally demonstrates that even though politicians

gain approval and liking by apologizing, an apology causes the politician to lose

status as measured by respect or willingness to re-elect.

Status loss could be modeled with a renegotiation proof contract based using

two-dimensional type. Up until now, the point of an apology was to shift the

principals attribution of the cause of a bad outcome from an internal and

controllable quality of the actor, to an external and uncontrollable quality of the

environment. Lee and Tiedens (2001a) find that even a successful shift of attribution

may not be good for the agent, if the agent is expected to have control over the

environment. I construct an equilibrium where a principal can offer a menu of tasks

that depend on both intentions and ability. I find that agents who apologize should

receive new tasks correlated to intention, while agents who do not apologize should

receive new tasks correlated to ability.

The details of these mechanisms along with a model of empathy-based partial

apologies based on information partitions are relegated to the appendix.


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4 Discussion: Experimental EvidenceIn the next two sections, I consider the implications of the models presented

first in the context of an experiment that I ran, and then in an application, the use of

apologies in medical malpractice.

4.1 Experimental EvidenceThe theory presented above constructs a rational model of apologies. The intent

is to understand apologies in broader contexts such as corporate culture or medical

malpractice litigation. Thus, it is useful to validate the models predictions with

experimental evidence in order to gain more confidence in the theorys validity.

Experiments also allow the exploration of additional regularities of behavior,

specifically other features of the dynamics of apologies, implications for social

welfare, and the effect of participant demographic characteristics.

It should be noted that a better test of the theory would have a more

sophisticated null hypothesis than apologies have no effect. The problem is that

alternative theories are not well specified. The Freudian notion that one apologizes

when he or she feel guilty is almost tautological. As for attribution theory, the

theory presented here is designed as a formalization and refinement of attribution

theory and thus would not yield differentiating predictions.

4.2 Background: The Trust GameThe trust game provides a natural test bed for the theory laid out. It is a two

player game where a task must be performed by one playerthe agentfor the

benefit of the otherthe principal. It is a game where responses differ across

individuals, and the actions of some individuals have the natural interpretation as

being nicer than the actions of others, a situation that lends itself to apologies.

Furthermore, the structure of the trust game provides clean measures of the five

main variables in the model: type, prior, outcome, apology and posterior.

The trust game is also useful because the typical behavior observed does not

conform to traditional restrictive models of rational behavior. The application of the

apology model to the trust game demonstrates the generality of the apology model,

and how rational models can be applied in such settings.

The trust game was first proposed by Berg, Dickhaut and McCabe (1995). The

canonical trust game is a one shot game with two players, a Principal/Investor (Paul)

and an Agent/Trustee (Amy). Paul is given some number, typically, ten tokens. Paul

can choose to keep any number of these; the rest gets invested with Amy, who


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receives three times the number that Paul invested. Amy then has the choice of

keeping any number of the tokens, the rest are returned to Paul. See Figure 3.

(Insert Figure 3 here)

In the Berg et al. (1995) study, the modal response was for the Principal to send

half, or five, tokens to the Agent, giving her a resource pool of 15 tokens to work

with. The modal response of the Agent was to return five back to the Principal,

providing an average net return on investment for the principal of 0%. The Berg et

al. (1995) study spawned a large literature that shows there is variation in behavior

by culture and government type (Koford, 1995; Ensminger, 2000), by gender

(Chaudhuri and Gangadharan, 2002), and by framing (Van Huyck, Battalio, and

Walters, 1995), thus the type of the person one interacts with matters. Also, there

is some debate as to whether the trust game represents reciprocity or altruism

(Gneezy et al. 2000; Cox, 2002; Chaudhrui and Gangadharan, 2003). My modelholds for either case.

There is a working paper by Schweitzer, Bradley and Hershey (2004) that

analyzes directly the impact of apologies in a repeated trust game scenario. Their

treatment uses deception and uses an actor in the role of the agent. The advantage of

my experiment is that I have observations for both the actions of the principal and

the agent. Furthermore, the game is constructed so that theoretical predictions can

be made in advance, and therefore the model can be directly tested.

Notably, however, the Schweitzer et al. (2004) analysis provides empirical

confirmation for the fool me once prediction from the dynamic cheap talk

model in the appendix. They find that though an apology does restore trust, a failure

after an apology is punished especially severely.

4.3 Experimental DesignThe experiments in this paper use a repeated and noisy variant of the trust game

programmed using z-Tree (Fischbacher, 1999). In each session, there are 8-12

subjects, with exactly half assigned the role of principal and half assigned the role of

agent (See Appendix B for the instructions). Each principal is randomly paired with

another agent with whom he plays a noisy trust game for ten periods. After each

period, each player learns the realized payoff for both players, and the game repeats.

After ten periods, that particular treatment ends. Each principal is re-matched with a

different agent from the same session until all possible pairings are made.

Given prior literature, I did not expect, nor did I find that the finite and certain

number of periods leads to unraveling. Indeed, the lack of unraveling in a setting


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where theory would traditionally expect it highlights an important feature of the

apology theory: plain vanilla utility based only on monetary gain is not assumed.

The noisy variant of the trust game used is similar in structure to the standard

form. The principal is given 10 tokens of which he can choose any integer number

greater than zero to keep for himself. The rest are entrusted to the agent. As in thestandard game, the agent receives the number entrusted by the principal, multiplied

by three which goes to form the agents resource pool. The agent can also choose to

keep any number of tokens for herself, with the remainder being returned to the

principal. However, instead of being directly returned to the principal, the noisy

variant introduces uncertainty. For each token that the agent returns, the principal

gets a cumulative five percent chance of project success. A successful project

yields the principal twenty tokens. Thus if 15 tokens are returned, the principal gets

twenty tokens with probability 0.75. Returning more than twenty does not help.

Since the return of more than 15 tokens was never observed in past experiments, the

game is essentially the same, in expectation, as the canonical trust game, except that

the principal is unaware of the number of tokens the agent actually returned.

At the end of each period, the agent is told the outcome of the project, and

given the opportunity to send a message. The only message she is allowed is a

message that reads I am sorry.20 Before each treatment, the players are all told the

communication cost associated with sending a message during that treatment.

Communication costs were chosen randomly for each player-pair and each

treatment. I varied the communications between 0, 5, 10, and 15 tokens.

(Insert Figure 4 about here)

The screens used in the experiment are found in Figure 4. The first and fourth

are seen by the principal. The second and third are seen by the agent. Both players

are told the payoffs for both players at the end of each period before the next begins.

Subjects were recruited through the Stanford Graduate School of Business

Behavioral Lab. Advertising for the experiment was targeted to those who had never

participated in an experiment before. Of the 58 subjects recruited for this study, 53%

had never been in an experiment involving deception. 57% were female. 44% were

white, 20% were Asian, 26% were other. 55% had never taken a psychology class

and 50% had never taken economics. 22% thought that deception might have been

involved. Regressions using these demographic data showed no significant effect on

observed behavior except non-white, non-Asians agents returned 17% fewer tokens,


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agents who had taken economics returned 11% more, and principals who had been

in experiments before entrusted on average 1.5 tokens more.

Subjects were given an instruction sheet (see Appendix B) that detailed all

aspects of the experimental design and encouraged to ask questions. They were

assured that the norms of economics precluded deception from being used. A trialrun was conducted in which the experimenter was available to answer questions.

Principals and agents were seated in separate rooms, to promote anonymity.

4.4 Correspondence to TheoryThis noisy variant of the trust game was chosen because it conforms well to the

model, and the measurable actions provide good proxies for the variables in the

model. Recall the five key variables of interest. Wherever averages are specified, I

consider averages for a given player over the ten period treatment, as well as

averages over the entire session.

: The theory defines as the measure of type where higher types yield

higher payoffs for the principal. In the trust game, two natural measures

have this property: the average number returned by the agent and the

average percent returned.

p: The theory definesp to be the prior belief the principal has that the agent

is a good type. How much the principal chooses to entrust to the agent is

monotonically increasing in p. A principal who is just maximizing tokens

would tautologically like to give more tokens to a higher type, because that

is how higher type is defined. Thus conversely, the more tokens a principal

entrusts, the higher his beliefs.21

y: This is the output for the principal. It is 20 in case of project success, and

zero in case of project failure.

a: In the theory, a=1 if the agent apologizes to the principal after a failure

and a=0 otherwise. This is simple to measure given my setup.

b: This is the principals posterior belief. Since the principals beliefs are

measured by how much he entrusts, the beliefs at the end of a period can be

measured by how much he entrusts at the beginning of the next.

20 I would have liked to allow more open ended messages, but the available software makesthis unworkable.21 Mathematically, a simple functional form would have be the percent an agent is

expected to return. Then a per-period profit would be xEx 3+= )10( , so

)(]13[/ bx = , so the principal should entrust everything for beliefs sufficiently


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It should be noted that it cannot be the case that the agent is merely maximizing

tokens, otherwise standard backward induction unraveling results would be

expected: no tokens would ever be entrusted, so no tokens would ever be returned.

Since unraveling rarely occurs in the trust game, there must be some other element

of the utility function so that at least some types (even an fraction suffices) willreturn a positive value. One such set of types that would generate such results would

be an fraction of a crazy type as in Kreps, Milgrom, Roberts, Wilson (1982).

Alternatively, type could reflect differences in discount rates, or levels of altruism.

Regardless of what type reflects, the theory presented here encompasses all of these

examples and requires only that types can be ordered by how much they return.

Finally, I check to make sure the existence conditions in Proposition 2 are

satisfied. To keep things simple for the subjects, constant costs of communication is

assumed,22 so what is required is the standard single crossing condition. Whether

single crossing actually holds depends on functional form assumptions. However,

from Proposition 2, if apologies are observed, then the condition must hold

presuming the other assumptions are also satisfied.

4.5 ResultsThe basic results reflect six sessions totaling 58 subjects, or 125 pairings of ten

periods each. All of the main predictions were found significant at the 95% level.

4.5.1 Basic Results


Consider first Table 2 which lists summary statistics of the relevant variables

broken down by communication costs. Apologies were used quite frequently, 43%

when costs were zero, 10% when the cost was five tokens, and not at all when the

cost was fifteen tokens. On average, of the 10 tokens that the principals received

each period, they entrusted about half in line with past trust experiments. From the

theory, one would expect principal profits to be maximized for intermediate costs

when apologies were most used. Profits were significantly higher at intermediate

communication costs in the pilot and in the majority of sessions, but two outlying

sessions negated that finding for the whole sample.

high, and nothing for beliefs sufficiently low. Though adding risk aversion would giveintermediate results.22 To be precise, I am assuming constant observable costs of communication. I interpretnoise in the estimation procedure to come from some internal psychic/social cost of apologythat I cannot observe.


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The remainder of this section is motivated by Proposition 1 of the theory,

which makes nine predictions about the anticipated relationship between the

variables, as summarized in Table 1.

First, the principals posterior belief, b, should be increasing in each of the four

other variables. The principal should entrust more in the following period if in theprevious period, the agent apologized, a, the project succeeded, y, the agent is a

better type, , or the principal had high beliefs and entrusted more to begin with, p.

Table 3 shows that these predictions are all significantly borne out.23 The addition of

agent fixed effects strengthens the results.


The regressions reported here include all observations from the first nine

periods and include period fixed effects. Separate regressions by period yields

consistent and similar results but given the limited number of observations, thesignificance is reduced for some coefficients.

The models predictions for the likelihood of apology are a bit more involved.

Higher types are expected to apologize more. Recall that the proxy for higher types

are those agents who over the entire session return a higher percent of their resource

pool. As for the effect of outcome on whether an agent apologizes, agents are

unsurprisingly more likely to apologize when there is a failure. The theory also

predicts that for extreme failures, the agent is also unlikely to apologize, but since

project outcome is a binary variable, this was not observed. The theory also predicts

that apology frequency is maximized for intermediate priors. For intermediate

priors, apologies have the most value since there is most uncertainty. Failure when

trust is low or when trust is high is not worth the cost.

As for predictions about project outcome, y, higher types yield higher project

outcomes, which is true by definition. As for the probability of success, it is also

maximized for intermediate values ofp. If the principal trusts the agent too much,

the agent has little incentive to put in high effort. The interior maxima predicted

by the theory for the effect of priors on apologies are borne out by the data. The

regression including fixed effects predicts the likelihood of apologies to be

maximized when the principal entrusts 6.07 tokens to the agent.


23 The tables are aggregated over the first 9 rounds of the game. The results still hold whenconsidering any one round, though for rounds 9 and 10, they become insignificant as end ofgame effects interfere. F-tests cannot reject the null hypothesis that period dummies for thefirst eight periods in the regressions in Table 3 and Table 4 are all the same.


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Note that both the tendering of an apology and the outcome of the project are

binary variables. The regressions were replicated using logit and probit analyses

yielding similar results. Also, redoing all presented regressions clustering standard

errors by principal or agent does not substantively affect significance.

4.5.2 Welfare and Dynamics

It was noted earlier that profits for principals are maximized at intermediate

apology costs. Though this holds and regressions are significant for four of the six

sessions, outliers in the other two sessions negate this for the sample as a whole.

In a preliminary analysis, there is some evidence for the dynamic predictions of

the model (see Appendix: Section 7.1). There is weak confirmation for the finding

of Schweitzer et al. (2004), that apologies followed by failure should be punished

more harshly than non-apologies followed by failure. Looking at the level of trust in

trials where communication costs are zero, I find some evidence for the converse ofthe fool me once, shame on you cheap talk equilibrium from Appendix A (see

Equation 38). Successes following non-apologies are rewarded especially highly, to

compensate agents who took an initial loss by not apologizing. However, these

results are not statistically significant.

4.6 Robustness ChecksTo check whether apologies are forward looking and economically motivated

by future payoff, or instead backward looking and psychologically motivated by

guilt, I look at what periods apologies are used in. Apologies show a significantly

negative time trend; the frequency of apology declines by 1.2% (t-value equal to

3.01) each period, where the average frequency is 21.5%.

(Insert Figure 5 here)

Also, if apologies are forward looking, then the agents continuation value

defined as the agents profits for the remainder of the game at the end of each

periodshould justify the cost of apology. Continuation values are consistently

higher for agents who apologize than for those who did not. Regressing continuation

value on the number of tokens an agent spends on an apology (including period

fixed effects) each token spent yields the agent 1.37 tokens future profit; this result

is robust to adding controls for prior beliefs and agent fixed effects.


Another backward looking hypothesis is that the agents are burning money in

order to restore fairness. However, a fairness motivation implies that higher agent


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profits should increase the likelihood of an apology. Regressions show fairly precise

insignificant results. Regressing difference in profits, either per period or

cumulative, on how much money is burnt, provides significant but very small

results.

(Insert Table 6)

Xiao and Houser (2005) run an ultimatum game experiment where cheap

messages with affective content do change the behaviors of the players. To explore

this possibility, I conducted the noisy trust game again. Instead of restricting the

message to I am sorry, I instead allowed a choice of five messages: I am happy,

I am sad, I am neutral, I am sorry, and I am angry. Each of the messages

except for I am neutral which was essentially never used, were used with about

equal frequency, though overall, messages were used much less frequently, about a

third as often as in the original game. Uncontrolled regressions show that othermessages like I am happy and I am sad have similar effects on beliefs as I am

sorry which continues to positively impact trust. However, once agent type is

controlled for using average number returned, one finds that only I am sorry is

significantly effective.24 I would argue that though the theory stands the same,

regardless of message, the specific I am sorry message triggers context cues in the

participants to play the separating equilibrium while for other messages the default

response is to pool. Alternatively, equilibrium play in the apology game is governed

by certain heuristically adapted strategies that are not triggered by other messages.

A third possibility is that there are unobserved costs associated with the words I am

sorry representing other proposed mechanisms such as commitment25 or status

loss, as modeled in the appendix. Clearly, there is something outside the context of

the model that affects behavior. Though rational signaling can explain much of the

variation, behavioral factors still play a role.


24 Another trial where messages such as I am happy or I am sad are assigned to playersyielded similarly insignificant results.25 The data is consistent with the fool me once model of commitment in the appendix,but not significantly so.


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5 Discussion: Applications

5.1 Apologies and the LawOne of the few areas of scholarly research that examines intensively the

question of apologies is in the area of law. Apologies have an important impact onthe outcome of cases. Unsolicited apologies can have an impact on conviction rates,

as well as sentence and judgment sizes (Rehn and Beatty, 1996). Furthermore, even

court ordered apologies appear to mitigate punishment (Latif, 2003). Yet as a result,

many legal scholars (Cohen, 2002; Taft, 2000) worry that by using apologies the

courts are interfering with a natural process of social remediation. This concern

has prompted lawmakers to consider legislation that exempts apologies from use in

the courts. This section examines the implications of the model for the legal system.

The model provides several predictions about the impact of apologies in legal

settlements. Consider sentences that mandate apologies. Since apologies are costly,

one would expect mandated apologies to be accompanied by lower sentences and

smaller fines. The model also predicts that apologies have higher impact when there

is more uncertainty regarding agents type. Apologies should have a larger effect on

sentences and fines for cases where the observables of the case are the least

predictive of the outcome. Cases that do go to trial are the most uncertain, and so

apologies should be more common in cases that go to trial rather than those that are

settled outside of court.

Consider cases where an apology is voluntarily tendered. If an apology is

considered a partial admission of guilt, then one would expect voluntary apologies

to be associated with higher conviction rates but lower sentences, presuming the

court would give leniency for good intentions.

The consequences of apology are of special concern in the area of medical

malpractice. Doctors are typically told to avoid admissions of fault and apologies in

particular, because of the risk of lawsuits (Pinkus, 2000; Novack et al., 1989). Yet

patient lawsuits in the event of error are rare, and only occur when the patient has a

poor relationship with the doctor (Gallagher et al., 2003; Huycke and Huycke,

1994). A lack of apology often is the reason why a lawsuit is even filed (Hickson etal., 1992; May and Stengal, 1990). Thus the lack of apologies often results in a

vicious cycle that recent Im sorry legislation hopes to break (Cohen, 2003).

Sixteen states ranging from Massachusetts, Texas and California have passed or are

considering legislation that exempts expressions of remorse from being used as

evidence in civil cases.


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A quick implication for the effect of this legislation is that in states where Im

sorry laws have been implemented, a voluntary apology cannot be used as evidence

of guilt, but potentially gives information about intentions. If intentions matter in

determining conviction, then voluntary apologies should then be associated with

lower conviction rates, relative to voluntary apologies in other states.The Im sorry laws can be modeled as an exogenous shift in the cost

function. Many of the proposed legal reforms reduce the cost of apologies in order

to increase their use. Though the effect depends on the shape of the reduction, it is

useful to consider the case where the cost is a constant function: =),(c .

A decrease in the cost of apologies, , will increase the likelihood of apology,

but it will also decrease the benefit, ),0(),1( ybyb , that an apology induces.

Apologies will have less impact, potentially counteracting the reform. The

attenuated impact of an apology is reflected in moral arguments made by Taft

(2000) that apologies have been subverted by the legal system. Taft argues that if

laws exempt apologies from the legal system, apologies would lose the moral

weight that comes from the associated responsibility an apology entails.

Furthermore, a decrease in the cost of apology will reduce the costs associated

with failure, thereby decreasing the incentives in the action stage, and increasing the

equilibrium probability of failure. This increase in moral hazard is echoed by Cohen

(2002) who worries that the predicted decrease in law suits filed will have a

detrimental impact on the natural process of remediation. Already, very few cases of

medical malpractice come to trial (Huycke and Huycke, 1994). One could argue thatsince these lawsuits are essential for restorative justice and efficient monitoring,

welfare would be enhanced if there were more lawsuits, not fewer.

Another testable implications, in states where such laws have been passed,

apologies should be more common, but at the same time less effective. Thus the

fraction of patients that pursue lawsuits given an apology should increase.

A full empirical analysis of the effects of the apology legislation on medical

malpractice payments that shows a sizable reduction in payments due to the laws

can be found in my other work (Ho, 2006).

6 Conclusions and Future ResearchApologies are a social institution that can be understood using the standard

rational economic toolkit. This paper provides a model that encompasses many of

the myriad settings in which apologies occur. In each setting, an apology acts as a

signal of the apologizers fitness for future interaction. Conditions are established


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for the unique existence of such equilibrium. Credence can be given to these

theoretical predictions because findings from a trust game experiment match the

theorys predictions better than alternatives.

For future research, the theory of dynamic signaling explored here could be

generalized. Additionally, the model can be specialized to applications in legalsettings and corporate culture. The apology model can be expanded with a better

notion of forgiveness and a larger community of actors. Also, the implication of this

model on the cultural evolution of emotions such as shame or guilt can be explored.

Formal empirical tests in real world contexts are possible from court cases, political

apologies, corporate scandals or surveys of corporate culture.

7 Appendix A Models of Cheap Apologies

7.1 Contracting with Commitment: Social ContractsIm sorry Ill never do it again

One approach to modeling cheap apologies is to take a contracting approach where the principal can commit to future termination strategies, creating a mechanism for ensuringtruthful revelation. In the other sections, Markov perfection limits the principal to choosingthe most attractive agent. Here, I proceed with the same model from section 2, except herethere is no explicit cost of apologies, and I relax the solution concept to a Markov PerfectEquilibrium that uses both the principals beliefs and the agents apology in the prior periodas the state variable. I also allow the principal to commit ex ante to a mixed strategy ofretaining the agent or not as a function of agents apology and outcome. I then look for aseparating equilibrium where good types always apologize and bad types never do.

Once again, agents produce an output and then choose whether to apologize or not.Principals now, instead of merely choosing between {continue, terminate}, now can choose

),( 1 ttt ya , the probability of termination, as a function of the apology in the previous

period, and the current period outcome.26In a separating equilibrium, beliefs as a function of apologies would be

(30)0),0(

1),1(

=

=

yb

yb

To implement such an equilibrium, the principal chooses ),( ya to arrive at a payoff

function, ),( bv , such that the following incentive compatibility constraints are satisfied:

(31)),1(),0(

),0(),1(

BB

GG

vv

vv

One way to obtain such a payoff function in the first stage of a two-period game is tohave appropriate payoffs in the second stage. I call this a fool me once, shame on you, foolme twice, shame on me contract. The intuition is that the principal would like to know theprivate information of the agent, but the agent has incentive to misrepresent her type; anapology is a claim to be a good type. However, if the agent claims she is a good type, the

26 It would also be sensible to allow t to depend on at, but I do not to simplify the analysis.


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principal will demand much more out of the agent, and tolerate failure far less, whereas ifthe agent does not apologize, the principal will be more forgiving of failure.

To simplify the problem, assume there are only two possible outputs for the principal

so that },{ yyy .27 Define the following:

(32) })),(:({]),(Pr[

})),(:({]),(Pr[

yyFyys

yyFyys

BB

B

GG

G

====

====

The utility of the agent is given by:

(33) )()),(,(),(),(),( 2121 vyauuaU ++=

Then the agents IC constraints so that only good types apologize are

(34))]()),(,1([)]()),(,0([

)]()),(,0([)]()),(,1([

22

22

BBBB

GGGG

vyEvyE

vyEvyE

which can be simplified using equation 32 and rearranged to yield

(35)B

B

G

G

s

s

yy

yy

s

s

1),0(),1(

),1(),0(

1

Moral hazard concerns provides two more constraintsan agent in the second roundmust get higher payoffs for success than failure:

(36)),0(),0(

),1(),1(

yy

yy

>

>

Combining these constraints gives us the following ordering of ),( ya :

(37) ),1(),0(),0(),1( yyyy >>>

Effectively, the marginal benefit of success in the second stage in the case of anapology in the first, must be higher than the marginal benefit in case of no apology in thefirst. An apology will lead the principal to believe the agent is a good type, but he willexpect better performance from her in the future. In the unreduced form, this means highereffort for the agent after an apology.

Alternatively, the following ordering is also possible given the constraints:

(38) ),0(),1(),1(),0( yyyy >>>

I argue this ordering is unlikely by considering the principals maximization problem.

If the probability of the bad type succeeding is still relatively high, which is likely if theprincipal has difficulty differentiating between bad and good, then the principal would preferthe ordering given in Equation 37.

Given the above analysis, I now return to the case of Paul and Amy. Paul may tolerateone failure from Amy to show up on time, but given the repeated failure, he is forced to endthe relationship.

27 These results generalize easily to continuous . They are messier but similar forcontinuousy.


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This game could be extended to N-periods except that after the second period, there iscomplete separation making signaling uninteresting. To get around this, I relax the completepersistence of type. Let there be some probability of mutation:

(39)p

p

B

t

B

t

G

t

G

t

===

===

+

+

]|Pr[

]|Pr[

1

1

Then even though there is full separation each period, the two period equilibrium canalso serve as a Markov perfect equilibrium for an n-period game where the prior is reset to

eitherp orp after each period.

In any case, the problem with the contract presented in this section is that it is notrenegotiation proof. Once an agent has apologized, she has established herself as the goodtype. At that point, the principal would not want to end the relationship. To solve thisproblem I introduce the ability for the principal to offer different tasks.

7.2 Contracting without Commitment: StatusIm sorry; Im an idiot.

7.2.1 Introducing tasksOne reason why contracting is difficult for the principal in the previous case is the

limitation of the principals strategy space. In this section, I give apologies moresophisticated meaning by expanding the space of principal responses. Now, instead ofcontinue or terminate, I allow the principal to offer a menu of tasks. Let there be a set

Z of tasks for each period, each task defined by an ordered triple ),,( zzz , where the

discount rate z reflects how long before that task comes up again and z is the correlation

of the next task with the current task in the dimension and z is the correlation in the

dimension. Until now, I have assumed that is identical across periods, while wasdrawn independently, limiting the principals choice set for the next period to

(40) ( ) ( ){ }0,0,0,0,1,1 ======= termtermtermcontcontcontZ

In this section, I expand the set of tasks available to the principal to a larger set. As

before, represents an internal dimension or disposition, while represents an externaldimension or situation, but here I consider scenarios where the agent is expected to havesome control over her situation. Whereas before, an agent could excuse poor performance toa bad situation, doing so now would admit to a lack of control, which is also bad for theagent. Formally, the change in the model is that now, payoffs to the agent are based not just

on the principals beliefs about , but also the principals beliefs about as well.Returning to the example of Paul and Amy, let the base task be be on time for tennis.

Let represent how much Amy cares about Paul, and let represent how able Amy is atshowing up to events on time. Now, consider two other tasks that Paul might like fulfilled:

talk to at party, and be on time for job interview. One would expect that success at talkto at party would be correlated with how much Amy cares about Paul, but not be correlatedwith how good Amy is at showing up on time for things. Conversely, be on time for jobinterview might depend very much on Amys ability to show up on time for things, but notdepend on Amys liking of Paul. Thus one might imagine an equilibrium in which if Amy islate for tennis and apologizes, then Paul would be happy to talk to Amy if he sees her at aparty, whereas if she did not apologize, Paul might be more likely to recommend Amy for ajob opening that depended on her on time arrival for the interview.


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This expansion of the set of tasks available allows the principal to offer a renegotiationproof menu that allows cheap apologies to carry meaning even if the principal is not able tocommit to a contract.

7.3 Model DetailsThe setup of the game starts again with the base model, and once again, the cost of

apology is set to zero. The main change is that now, both and are semi-persistent, andthe degree of persistencei.e. correlation, across periodsis determined by the principalschoice of tasks. The agents choice of actions each period is the same as before, she produces an output for the principal, and then, upon realization of the output, decideswhether to apologize or not. The principal offers a menu of two future tasks, one if the agentapologizes, and the other if she does not. The choice of task determines the distribution that

governs the agents type ),( 11 ++ tt in the next period.

The principals payoffs are the same: maximize his output across periods.

(41) =t

ttP yU ),(

The agents payoffs per period payoff is also the same, except now, the agents

continuation value depends not just on the principals beliefs about her internal type, b , but

also the principals beliefs about her external type, b :

(42) ),,,(),( ttttA bbvuU +=

In a two period game, the agents second period payoff will depend on the task selectedby the principal at the end of the first.

(43) ),()),((),( 111 ++++= tttttttA uyazuU

I add a number of simplifying assumptions. None are crucial, but they make analysis

more comprehensible. Limit the external dimension to two values so that ),{HL

one of which represents low ability, and the other representing high ability. I retain),{ BG so that there are good agents who care about the principal, and bad agents

who do not care. Then, for any particular task in period t, the agents type is given by

),( tt which can take one of four values.

Now, assume also that there are only two possible outputs for the principal, }1,0{y ,a task can be a failure or a success. Assume also that an agent succeeds at a given task only

if she is both of good disposition, and high ability: ),( HG :

(44)

0),(

0),(

0),(

1),(

=

=

=

=

LB

HB

LG

HG

y

y

y

y

Similarly, assume that agents utility is increasing in type. Specifically, assume theagents utility for consumption is given as:


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(45)

0),(

0),(

0),(

1),(

=

=

=

=

LB

HB

LG

HG

u

u

u

u

Note that the agent is also only happy when she is successful. In this simplified form,there is preference alignment between principal and agent, when it comes to the task at hand.This alignment is not necessary, but it makes the conflict of interest introduced by taskassignment more apparent.28

Some notation will be helpful. Recall for a given task z to be assigned in the next

period, z is the correlation of the new 1+t with the current t , while z is the

correlation of 1+t with t . The principals prior that the agent isG for a given taskzis

zp and the prior that the agent isH for a given task is zq . Then, define:

(46)

)1()1(],|Pr[

)1()1(],|Pr[

)1()1(],|Pr[

)1()1(],|Pr[

11

1

11

1

zzqz

L

t

H

tz

zzqz

H

t

H

tz

zzpz

B

t

G

tz

zzpz

G

t

G

tz

qqqqqzq

qqqqqzq

pppppzp

pppppzp

z

z

z

z

====

+====

====

+====

+

+

+

+

I now look for a renegotiation-proof Markov Perfect Equilibrium in pure strategiesagain using beliefs and last period apologies as a state space. Given the stark production

technology specified in Equation 44, if the principal observes a success, 1=y then he

knows for sure that the agent is ),( HG and seeks to assign a task as similar to the

current task as soon as possible. That is, a task where ,, are all close to one. If Amy

shows up to tennis on time, then Paul will ask her back to play tennis again the followingweek.

In the event of a failure, 0=y , the set of possible agent types narrows to

)},(),,),,{( LBHBLG ( . Assuming that success is relatively common so thatboth p and q are relatively high, the third case would be rare. Consider a strategy by the

principal that allows him to distinguish between the first two cases.The principal offers a menu of two tasks after a failure. An agent who apologizes gets

task 1z , and an agent who does not apologize gets task 0z . To get separation, the principal

selects tasks so that ),( LG types apologize and ),( HB types do not. The incentive

compatibility conditions are:

(47)111000

000111

)()()()(

zzzz

zzzz

qpzqpz

qpzqpz

Ideally, the principal would offer one task perfectly correlated on the internal

dimension, )0,1,1( 111 === zzz , and the other task perfectly correlated on the

28 As before, one could introduce moral hazard via a hidden action for the agent that wouldyield the same reduced form.


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external dimension, )1,0,1( 000 === zzz , but the actual assignment depends upontask availability. The conflict arises because payoffs for the principal are undiscounted. Iassume that the principal interacts with potentially many agents, and thus could have the taskfilled by another. Thus, the difference in correlations must be sufficient to overcome thedifference in discount rates. For example, in the example with Paul and Amy, the jobinterview task may come quite infrequently, and would have a particularly low discount

rate. If no appropriate task is available, then apologies would be uninformative. Often,agents apologize consequence free.

In the case of politics, as demonstrated by Lee and Tiedens (2001), an apol

Benjamin Ho Economics of Apologies

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