Team Reasoning

Natalie Gold, King’s College London

Formal Ethics, Bayreuth, July 2015

I: INTER-PERSONAL TEAM REASONING

TWO PUZZLES OF GAME THEORY:

Prisoner’s Dilemma and Hi-Lo

Two Puzzles of Game Theory I

The Prisoner’s Dilemma:

P2cooperate

defectP1 cooperate 3, 3 1, 4

defect 4, 1 2, 2

Two Puzzles of Game Theory I

The Prisoner’s Dilemma:

P2cooperate

defectP1 cooperate b, b d, a

defect a, d c, c

a > b > c > db > (a + d)/2

Two Puzzles of Game Theory II

Hi-Lo:

P2high

lowP1 high 2, 2 0, 0

low 0, 0 1, 1

Two Puzzles of Game Theory II

Hi-Lo:

P2high

lowP1 high a, a 0, 0

low 0, 0 b, b

a > b

SCHEMATA OF PRACTICAL REASONING:

Framework for representing modes of reasoning

Schemata of Reasoning

(1) There are no English mountains over 1000 metres high.

(2) Snowdon is a mountain which is 1085 metres high.______________________________________ Snowdon is not in England.

Schema 1: Individual RationalityP1

left right

O1 O2 O1 O2

(1) I must choose either left or right.

(2) If I choose left, the outcome will be O1.

(3) If I choose right, the outcome will be O2.

(4) I want to achieve O1 more than I want to

achieve O2.__________________________________________________________________________

I should choose left

Individually Instrumental Reasoning for P1 in Hi-Lo

(1) I am P1 in Hi-Lo.

(2) The probability that P2 will choose high is 0.5._______________________________________

I should choose high.

Collective ChoiceP1

left right

-----P2 -------------------------- P2-----left right left right

O1 O2

O3

O4

O1 O2 , O3 , O4

Schema 2: Collective Rationality(1) We must choose one of (left, left), (left, right), (right, left)

or(right, right).

(2) If we choose (left, left) the outcome will be O1.

(3) If we choose (left, right) the outcome will be O2.

(4) If we choose (right, left) the outcome will be O3.

(5) If we choose (right, right) the outcome will be O4.

(6) We want to achieve O1 more than we to achieve O2, O3 or O4.

_______________________________________We should choose (left, left).

Formalising Team Reasoning: Definitions

• A set S of individuals.

• Each individual has a set of alternative actions, from which she must choose one, A.

• A profile of actions assigns to each member of S one element of her set of alternative actions.

• For each profile, there is an outcome.

• A payoff function is a function which assigns a numerical value to every outcome.

Group Identification

Now consider:any individual i, and any set of individuals T, such that i is a member of T and T is a weak subset of S.

We will say that i identifies with T if i conceives of T as a unit of agency, acting as a single entity in pursuit of some single objective.

Common KnowledgeA proposition p is common knowledge in a set of individuals T if: (i) p is true; (ii) for all individuals i in T, i knows p; (iii) for all individuals i and j in T, i knows that j

knows p; (iv) for all individuals i, j, and k in T, i knows that j

knows that k knows that p; and so on.

Schema 3: Simple Team Reasoning (from a group viewpoint)

(1) We are the members of S.

(2) Each of us identifies with S.

(3) Each of us wants the value of U to be maximised.

(4) A uniquely maximises U.____________________________________Each of us should choose her component of A.

Schema 4: Simple Team Reasoning(from an individual viewpoint)

(1) I am a member of S.

(2) It is common knowledge in S that each memberof S identifies with S.

(3) It is common knowledge in S that each member of S wants the value of U to be maximised.

(4) It is common knowledge in S that A uniquely maximises U.____________________________________I should choose my component of A.

TEAM REASONING COMPARED WITH

INDIVIDUAL REASONING:

Team Reasoning Solves the Puzzles that Individual Reasoning Cannot

Team Reasoning solves Hi-Lo

• Assume payoff function U represents the welfare of the players in the group.

• The players’ payoffs are equal, so make the values of U equal to the players’ common payoffs.

• Then (high, high) is the profile that uniquely maximises U.

• Each player can use Schema 4 to reach the conclusion that she should choose high.

Team Reasoning solves the Prisoner’s Dilemma

• Assume that U treats the players symmetrically, denote payoff to (C, C) as uC, payoff to (D,D) as uD, payoff to (D,C) and (C,D) as uF (for ‘free riding’).

• Assume that U is increasing in individual payoffs, which implies uC > uD.

• Given the condition b > (a + d)/2, it is natural also to assume uC > uF.

• Then (C,C) is the profile that uniquely maximises U.

• Each player can use Schema 4 to reach the conclusion that she should choose cooperate.

Is Team Reasoning Necessary?Team reasoning vs. benefaction

A prisoner’s dilemma with transformed payoffs:

Let a = 10, b = 8, c = 6 and d = 0Let payoff function for the group {P1, P2} be the

average of the payoffs for the two

P2cooperate

defectP1 cooperate b, b d, a

defect a, d c, c

Is Team Reasoning Necessary?Team reasoning vs. benefaction

A prisoner’s dilemma with transformed payoffs:

Let a = 10, b = 8, c = 6 and d = 0Let payoff function for the group {P1, P2} be the

average of the payoffs for the two

P2cooperate

defectP1 cooperate 8, 8 0, 10

defect 10, 0 6, 6

Is Team Reasoning Necessary?Team reasoning vs. benefaction

A prisoner’s dilemma with transformed payoffs:

Let a = 10, b = 8, c = 6 and d = 0Let payoff function for the group {P1, P2} be the

average of the payoffs for the two

P2cooperate

defectP1 cooperate 8, 8 5, 5

defect 5, 5 6, 6

THEORIES OF TEAM AGENCY:

Why think as a team?

How are Teams Formed?

• Rationality/ morality

• Framing

• Mutual commitment

• Non-rational assurance

Team Reasoning When Not All Group Members Group Identify

• Let T be any subset of S (the team who identify)

• Each member of the team T identifies with S

• Each member of T wants U to be maximised

• U represents what people want as members of S

Team Agency Required by Rationality/ Morality

Morality Requires Team Agency

• Hodgson (Rule Utilitarianism)

• Regan (Cooperative Utilitarianism) ‘what each agent ought to do is to co-operate, with whoever else is co-operating, in the production of the best consequences possible given the behaviour of non-co-operators’ (1980, p. 124)

Schema 5: Restricted Team Reasoning

(1) I am a member of T.

(2) It is common knowledge in T that each member of T identifies with S.

(3) It is common knowledge in T that each member of T wants the value of U to be maximised.

(4) It is common knowledge in T that AT uniquely maximises U, given the actions of non-members of T.____________________________________I should choose my component of AT.

Rationality/ Morality Requires Team Agency

• Anderson (principles of self-identification)– Rationality: identity prior to rational choice (2001, p. 32).

– Morality: requires us to transcend our various identities and harmonise their demands, by identifying with a community that comprehends them all – the Kantian Kingdom of Ends

• Hurley (specify agent-neutral goals)‘survey the units of agency that are possible in the

circumstances at hand and ask what the unit of agency, among those possible, should be’; and‘ask ourselves how we can contribute to the realization of the best unit possible in the circumstances’ (1989, pp. 136-159)

Team Agency as the Result of Framing

Team Membership and FramingBacharach (2005)

• A frame is the set of concepts a player uses when thinking about her situation.

• In order to team reason, a player must have the concept ‘we’ in her frame.

• Frames can be primed, or induced

• Hi-Lo and the PD both have a property called strong interdependence

Framing and the Prisoner’s Dilemma

• Bacharach (2005)‘In a Prisoner’s Dilemma, players might see only, or most powerfully, the feature of common interest and reciprocal dependence which lie in the payoffs on the main diagonal. But they might see the problem in other ways. For example, someone might be struck by the thought that her coplayer is in a position to double-cross her by playing [defect] in the expectation that she will play [cooperate]. This perceived feature might inhibit group identification.’ (2005, Chapter 2, Section 4.2)

T-Conditional Common Knowledge

A proposition p is T-conditional common knowledge in a set of individuals T if: (i) p is true; (ii) for all individuals i in S, if i is a member of T, then i knows p; (iii) for all individuals i and j in S, if i is a member of T, then i knows that if j is a member of T, then j knows p; and so on for all individuals i and j in T, i knows that j knows p;

Schema 6: Circumspect Team Reasoning (1) I am a member of T.

(2) It is T-conditional common knowledge that each member of T identifies with S.

(3) It is T-conditional common knowledge that each member of T wants the value of U to be maximised.

(4) It is T-conditional common knowledge that A uniquely maximises U, given the actions of non-members of T.____________________________________I should choose my component of A.

Circumspection and the Prisoner’s Dilemma• Set S = {P1, P2}

• Let 0 < ≤ 1 for each individual be the probability that the ‘we’ frame comes to mind, in which case the individual identifies with {P1, P2}

• if this frame does not come to mind, the player conceives of himself as a unit of agency and thus, using best-reply reasoning, chooses the dominant strategy defect

• which protocol maximises U, given the value of ?

Circumspection and the PD cont.:Optimal profile when in the ‘We’ frame

Two cases to consider:

1) If uF uD, then (cooperate, cooperate) is the U-maximising protocol for all possible values of

2) If uD > uF, which protocol maximises U depends on the value of - at high values, (cooperate, cooperate) is uniquely optimal- at low values, (defect, defect) is uniquely optimal

Framing is a Psychological Mechanism

• If uF uD or the value of is high enough to make (cooperate, cooperate) the uniquely optimal protocol, we have a model in which players of the Prisoner’s Dilemma choose cooperate if the ‘we’ frame comes to mind, and defect otherwise.

• For any given individual, if she identifies with S and wants U to be maximised, it is instrumentally rational for her to act as a member of T.

• The parameter is interpreted as a property of a psychological mechanism

Team Agency Produced by Commitment

Commitment and Team Membership

• Gilbert (1989) ‘plural subject’ - formed by public expression of commitment

• Hollis (1998) ‘most remarkable change in man’ - collective act of commitment in Rousseau

Commitment and Framing Accounts Compared

• Bacharach: cannot choose your frame, but opportunities for cooperation will tend to prime a we–frame

• Gilbert: create plural subject with the rational intention of arriving at mutually beneficial solutions to problems of coordination or cooperation

• Bacharach’s concept of framing allows one person to choose an action with the intention of affecting someone else’s frame.

• ‘Would you like to dance?’

Gilbert: the first stage in a process which may lead to a common understanding that the two people are a plural subject in relation to a dance.

Bacharach: part of a process by which two people influence one another’s frames.

• Schema 4 is compatible with Gilbert.

Schema 4 and Commitment

• On this interpretation schema 4 is not one of instrumental reasoning.

• The rationality of acting as a member of a team derives from the rationality of fulfilling one’s commitments or intentions.

• There is no problem that S may be different from T.

Team Agency and Assurance

Assurance and Team Reasoning• Schemas 4, 5 and 6 yield unconditional conclusions

• Schema 4 has assurance - only tells the agent to choose her component of the best profile in situations in which it tells the other members of S chose theirs- there is common knowledge in S that everyone identifies with S

• This does not carry over to Schemas 5 and 6:- each member of the team T identifies with S- each member of T wants U to be maximised - U represents what people want as members of S

Logic of Team Reasoning

• Sugden (2003) presents a logic of team reasoning.

• A ‘logic’ is an internally consistent set of axioms and inference rules, no agent neutral concepts of ‘validity’ and ‘rationality’.

• Represents team reasoning as a particular inference rule which, as a matter of empirical fact, many people endorse.

• Re-interpret Schema 4 as an inference rule.

• But what about assurance?

Cross-Personal Reason to Believe

There is cross-personal reason to believe a proposition p is in a set of individuals T if:

(i) for all individuals i and j in T, where i j, i has reason to believe that j has reason to believe

p;

(ii) for all individuals i, j, and k in T, where i j and j k, i has reason to believe that j has reason to believe that k has reason to believe p;

and so on

Schema 7: Mutually Assured Team Reasoning

(1) I am a member of S.

(2) In S, there is common cross-personal reason to believe that each member endorses and acts on mutually assured team reasoning with respect to S and U.

(3) In S, there is common reason to believe that A uniquely maximises U.

(4) I endorse mutually assured team reasoning.____________________________________I should choose my component of A.

Team reasoning in GroupsTheories of team reasoning differ on several dimensions:

• how to deal with cases in which not every member of the relevant group can be relied on to identify with the group

• whether group identification is a product of psychological framing or conscious commitment

• whether each individual’s engaging in team reasoning is conditional on assurance that others engage in it too

• if so, whether assurance is generated by common knowledge of the psychology of framing, by joint commitment, or by experience.

But team reasoning is just as coherent and valid as the best-reply reasoning of conventional game theory.

I: INTRA-PERSONAL TEAM REASONING

Decision theory and dynamic choice

“The individual over time is an infinity of individuals” (Strotz, 1955-6)

Dynamic Choice

• at time t, the decision is made by a distinct transient agent, the person at time t

• self-control as a problem of diachronic consistency: – the transient agent in a early period would like to

implement a particular series of action but she knows that when the transient agents of later periods come to make their choices, they will not have incentives to carry out their part of the plan.

Backwards Induction

Your corn is ripe to-day; mine will be so to-morrow. ‘Tis profitable for us both, that I shou’d labour with you to-day, and that you shou’d aid me to-morrow. I have no kindness for you, and know you have as little for me. I will not, therefore, take any pains upon your account; and should I labour with you upon my own account, in expectation of a return, I know I shou’d be disappointed, and that I shou’d in vain depend upon your gratitude. Here then I leave you to labour alone: You treat me in the same manner. The seasons change; and both of us lose our harvests for want of mutual confidence and security.

(Hume, 1739-40/1978, pp. 520-521)

Reason by backwards induction

• Two transient agents, P1 and P2

• P1: what will P2 do if I go to the middle?

• Concludes ‘I Should go to the middle’

Reason by backwards induction

• Two transient agents, P1 and P2

• P1: what will P2 do if I go to the middle?

• Concludes ‘I Should go to the middle’

Reason by backwards induction

• Two transient agents, P1 and P2

• P1: “What will P2 do if I go to the middle?”

• Concludes ‘I Should go to the middle’

Reason by backwards induction

• Two transient agents, P1 and P2

• P1: “What will P2 do if I go to the middle?”

• Concludes: “I should go to the middle”

Present bias: ‘Special contiguity’

In reflecting on any action, which I am to perform a twelve-month hence, I always resolve to prefer the greater good, whether at that time it will be more contiguous or remote; nor does any difference in that particular make a difference in my present intentions or resolutions. ... But on my nearer approach, those circumstances, which I at first over-look’d, begin to appear, and have an influence on my conduct and affections. A new inclination to the present good springs up, and makes it difficult for me to adhere inflexibly to my first purpose and resolution.

(Hume,1739-40/1978, p. 536)

Strategic interaction between selves

Strategic interaction between selves

• Saturday night at the movies:

Strategic interaction between selves

• Saturday night at the movies:– Week 1, a mediocre movie– Week 2, a good movie– Week 3, a great movie– Week 4,

Strategic interaction between selves

• Saturday night at the movies:– Week 1, a mediocre movie– Week 2, a good movie– Week 3, a great movie– Week 4,

• When do you write the report?

Strategic interaction between selves

• Saturday night at the movies:– Week 1, a mediocre movie– Week 2, a good movie– Week 3, a great movie– Week 4,

• When do you write the report?

Strategic interaction between selves

• Saturday night at the movies:– Week 1, a mediocre movie 3– Week 2, a good movie 5– Week 3, a great movie 8– Week 4, 13

• When do you write the report?

Solutions

• Alter opportunities (destroy options, pre-commitments)

• Alter payoffs/ incentives

• But NOT intentions/plans

Parallel between groups and selves:

Externalities and internalities

Pollution as a Prisoner’s Dilemma/ Externality

• Externality: An agent’s action has costs/ benefits that are not completely captured by the agent

• There is a cost to refraining from polluting• There are benefits to everyone when the

individual refrains• For any individual, the cost may outweigh the

individual benefit• But everyone prefers the situation where no-one

pollutes to that where everyone pollutes

Giving up Smoking: a sequential prisoner’s dilemma/ internality

• But transient agent gets enjoyment out of smoking a cigarette, which she would forgo if she refrained from smoking (a cost)

• Benefit (reduced risk of cancer) is an externality - not entirely captured by transient agent

• Transient agent rationally “defects”, i.e. smokes• Transient agent thinks that it is better, over her

lifetime, to be a non-smoker rather than a smoker, as that reduces the risk of cancer

Utility: experiences, preferences, welfare

• Decision utility. The utility as inferred from decisions. This is revealed preference.

• Experience utility. The pleasure and pain experienced at a given moment in time (instant or total).

A Simple Model of Dynamic Choice(including intra-personal altruism)

• One person, three periods• In each period t, the individual’s consumption is xt • For each period t, an experienced utility function vt

• For each period t, a lifetime utility function ut

ut = βt, 1 v1(x1) + βt, 2 v2(x2) + βt, 3 v3(x3)

βt, r > 0 for all t and r

Example: Jo’s Examination

• In period 3, Jo takes an examination. • In each of periods 1 and 2, she can choose between revising for

the examination (work) and relaxing (rest). • The experienced utility of working is –3, that of resting is 0. • In period 3, experienced utility is

0 if Jo has rested on both previous days, 5 if she has rested on one day and worked on the other,

10 if she has worked on both. • In the lifetime utility function, the present experienced utility is

given twice as much weight as that of other periods.

Jo’s Examination cont.

Jo2

workrest

Jo1 work 1, 1, 14 –1, 2, 7

rest 2, –1, 7 0, 0, 0

Note: the interaction between Jo1 and Jo2 is a sequential Prisoner’s Dilemma

Team Reasoning

“What should we do?”

Team Reasoning

“What should we do?”

Why Intra-Personal Team Reasoning?

(1) Indeterminacy of rational choice theory-challenge the backwards induction solution

(2) Philosophical arguments about agency- personal utilitarianism vs. agency

Personal Utilitarianism

• Parfit (1984) – Personal identity is Relation R – overlapping chains

of strong connectedness– No rational reason to care about future selves– But moral obligations to look after future selves– But personal identity is not what matters– “If I fail to identify with that earlier self, I am in

some ways thinking of that self as a different person” (Parfit Reasons and Persons, p.319)

Team Agency

• Korsgaard (1989) – deliberative standpoint from which you choose– standpoint of practical reason requires unity of agency – or we couldn’t pursue projects

• Cf. Frankfurt’s ‘Standpoint’ = how agent’s various attitudes, and other relevant features of her psychology, come together to constitute a coherent, relevant standpoint

• Bratman (2014): timeslice evaluative vs. cross-temporal standpoint

Re-cap

• Team reasoning leads to rational cooperation by people in groups

• Problems of self-control as intra-personal dilemmas

• Team reasoning leads to rational formation and following of intentions

Thanks!