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Bayesianism Without Priors, Acts Without Consequences

Robert Nau

Fuqua School of Business

Duke University

ISIPTA 2005

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Possible sources of imprecision in subjective (epistemic) probabilities

1. Preferences are incomplete or only partially revealed, so beliefs are represented by convex sets of probabilities.

2. Preferences are complete but violate the independence axiom, so beliefs are represented by non-additive or second-order or otherwise generalized probabilities.

3. Preferences are complete and satisfy independence, but beliefs are entangled with state-dependent utilities, so they are represented by risk neutral probabilities (marginal betting rates for money).

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Does source #3 matter? Yes!• Inseparability of probabilities and utilities is potentially

very problematic for game theory (common knowledge of utilities? common prior probabilities??)

• Also problematic for economic models which try to separate the effects of “information” and “tastes”

• Also problematic for characterizations of aversion to risk and/or uncertainty that refer to “true” expected values or “constant” acts as benchmarks

• And it’s also problematic for the measurement of previsions!

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How are previsions measured?

• De Finetti’s method: the lower prevision of x is the quantity of money you are willing to pay to receive $x.– But what if utility for money is nonlinear??

• Walley/De Cooman’s method: the lower prevision of x is the quantity of utility you are willing pay to receive x utiles.– But how are “utiles” measured, and what if utility is

state-dependent?

The “utile” problem cannot be finessed away, because...

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How are utilities measured?• The authority for a subjective scale of utility comes from

Savage (1954) and Anscombe-Aumann (1963).• Their models assume the existence of a set of primitive

consequences (“states of the person”).• Acts are arbitrary, often-counterfactual, mappings of

states to consequences (or objective lotteries thereon).• The utility of act x is defined as the number such that x

is indifferent to a lottery that yields the best and worst consequences with objective probabilities and 1.

• This definition depends on the conventional assumption that consequences have state-independent utility, but...

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State-independence of utility is inherently untestable

• It is unreasonable to expect that, in all situations, it will be possible to define “consequences” or “prizes” so that their utilities will be a priori state-independent—particularly in situations that people care about.

• The usual axioms of state-independence (Savage’s P3 or Anscombe-Aumann’s monotonicity) anyway do not rule out the possibility of state-dependent utility scale factors.

• Aumann’s 1971 letter to Savage: what if the event in question is the survival of your beloved spouse or the debunking of the theory on which your life’s work is based? (or the performance of your retirement portfolio...)

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This paper:• Axiomatizes a simple additive representation of preferences

without explicit consequences• The only primitives are discrete “moves” of two players—

the DM and nature—and monetary gambles• Implicitly every combination of a move of the DM, a move

of nature, and a monetary payoff, is treated as a distinct consequence

• Doesn’t yield “true” probabilities—but doesn’t need to!• Provides a sufficient foundation for Bayesian methods of

decision analysis and game theory based on no-arbitrage and risk neutral probabilities, which are marginal previsions (local betting rates measured in terms of money, not utiles)

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Modeling framework

• Let A = {a1, …, aI} denote a finite set of strategies (alternative courses of action)

• Let S = {s1, …, sJ} denote a finite set of states. Non-empty subsets of S are events.

• Let w = {w1, …, wJ} W J denote an observable allocation of money over states (i.e., a side-gamble)

• An act is a strategy-allocation pair (a, w)• An outcome is a triple (a, s, w)

This is essentially the Arrow-Debreu state-preference framework augmented by a finite set of strategies

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Preference axioms

Axiom 1: (Ordering) is a weak order, and for every strategy the restricted preference relation over allocations is a continuous weak order.

Axiom 2: (Strict monotonicity) w*j > wj (a, w*j wj) (a, w) for all a, j, w, w*j.

(More money preferred to less, and no null states)

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Axiom 3: (Strategic generalized triple cancellation) For every pair of strategies a and b, allocations w, x, y, z, state j, and constants wj*, xj*:

If (a, w) (b, x) and (a, w*j wj) (b, x*j x-j) and (a, wj yj) (b, xj,zj) then (a, w*j yj) (b, x*j zj).

In words: if changing wj to wj* is no worse than changing xj to xj* when the background is a comparison of (a, w) vs. (b, x) , then changing wj to wj* cannot be strictly worse than changing xj to xj* when different allocations y and z are received in the other states.

Axiom 4: (Overlap among strategies) There exist allocations {w(a), aA } in the interior of W such that (a, w(a)) ~ (b, w(b)) for all strategies a and b.

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Main result: general additive model

Theorem 1: Axioms 1-4 hold iff is represented by a cardinal utility function U having the strategy-dependent additive-across-states form:

U(a, w) = ,

in which {vas} are strictly increasing, continuous evaluation functions for money that are unique up to joint transformations of the form vas + s+as where >0 and, for every a, .

Note: preferences among strategies without any side-gambles are represented by {s vas(0)}.

)( sasSs

wv

0Ss

as

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Special cases of the general additive model

1. SEU with quasi-linear utilityvas(ws) = ps(uas+ws)

2. SEU with state-dependent linear utilityvas(ws) = ps(uas+sws)

3. SEU with state-independent utility and prior stakesvas(ws) = psu(ws+Ws)

4. General state-dependent SEU vas(ws) = psuas(ws)

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Which SEU preference parameters are theoretically observable ?

• The DM’s “true” subjective probabilities for all events are observable only in special case 1 (quasi-linear utility).

• True probabilities are distorted by state-dependence and/or effects of prior stakes in cases 2 and 3.

• True probabilities are irrelevant in special case 4 (hopelessly entangled with utilities).

• But… so what ! True probabilities are not needed for analysis of decisions (or games).

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What’s really observable via finite measurements?

• The decision maker’s preference order has an additive representation in terms of evaluation functions {vas}, which are infinite-dimensional vectors.

• What sort of finite-dimensional measurements of preferences can be performed to enable decision analysis & game-theoretic analysis?

• How can we tell if the decision maker is acting “rationally” based (only) on this information?

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Analysis of a finite decision problem

• Henceforth, assume that only a finite number M of concrete acts are available, accompanied by “small” side-gambles.

• W.l.o.g. let each concrete act be considered as a unique strategy with zero allocation, e.g., “strategy m” means act (am, 0) for m = 1, …, M

• An outcome of the decision problem is a pair (m, s).

• Let vms(w) henceforth denote the evaluation function for money in outcome (m, s).

• How can we predict the decision maker’s choice of m via credible elicitation of information about her “beliefs” and “values” encoded in {vms(w)} ?

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De Finetti’s excellent idea• Let beliefs (subjective probabilities) be revealed through

offers to accept small gambles.

• Let rationality be defined as the avoidance of arbitrage (“no Dutch books”).

• Under-appreciated virtue: this approach not only defines beliefs and rationality, it renders them common knowledge in a practical sense.

• It can also be extended to the revelation of values (albeit with imperfect separation of beliefs/values).

• It also provides a natural bridge to game theory and asset pricing theory

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Measurable parameters of belief: risk neutral probabilities

• The decision maker’s risk neutral (betting) probabilities given strategy m are the normalized derivatives of the evaluation functions {vms}, evaluated at w=0:

• Note that they are, in general, act-dependent

• Conditional on the “event” that strategy m is chosen, for whatever reason, z is a utility non-decreasing “belief gamble” iff 0 z · m E [z]

n

k ms

msms

v

v

1)0(

)0(

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Common knowledge of beliefs (CKB)

• Definition: vectors {zk} are acceptable gambles if the DM is willing to let an opponent choose small non-negative multipliers {k} and receive a total payoff of k kzk from the opponent

• Axiom CKB: for each m, the vectors z = ±(1s ms) are acceptable given strategy m, where 1s is the indicator for state s and ms is the risk neutral probability of state s given given strategy m.

• I.e., the DM is willing to buy/sell Arrow-Debreu securities on state s at price ms in the event she chooses strategy m.

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In what sense do risk neutral probabilities {m } represent “beliefs”?

• They coincide with hypothetical “true” probabilities (only) in the special case of quasi-linear utility

• In all other cases, they are strictly more useful (e.g., they suffice to determine marginal prices and risk premia)

• They are updated via Bayes’ rule on receipt of new experimental information, exactly like “true” probabilities

• Individuals ought to eventually agree on risk neutral probabilities, not “true” probabilities (given the opportunity to gamble with each other)

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Measurable parameters of value: “value gambles”

• In the event strategy m yields greater utility than strategy n, the gamble mn defined by

whose state-by-state increments of cardinal utility are proportional to the state-by-state differences in value between m and n, evaluated from the perspective of the local marginal values for money that apply under the chosen strategy m, is utility non-decreasing.

)0(

)0()0(

ms

nsmsmns v

vv

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In what sense do {mn} represent “values”

• They coincide exactly with vectors of utility differences in the special case of quasi-linear utility

• They are independent of beliefs (subjective probabilities) in the case of state-dependent SEU

• They are independent of information in the case of scientific experiments

• Axiom CKV: for every strategy m and alternative strategy n, the vector mn is an acceptable gamble.

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Decision analysis in terms of acceptable gambles

• By construction: m mn U(am, 0) U(an, 0).

• Hence the direction of preference between any two concrete acts can be determined from observations of acceptable gambles under CKB and CKV.

• This suffices to determine the optimal act, even though “true” probabilities and utilities have not been observed

• But there is also another way to determine the “rational” outcome of the decision problem...

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Common knowledge of rationality

Definition: There is ex post arbitrage in outcome (m, s) if there exists a non-negative linear combination of acceptable gambles whose total payoff to the decision maker[s] is non-positive in all outcomes and strictly negative in state (m, s)

Axiom CKR: There is no ex post arbitrage.

• In other words, if there is ex post arbitrage in outcome (m, s), then it should not happen.

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Solution of the decision problem “by arbitrage”

• Fundamental theorem of decision analysis [one DM vs. nature]: Given axioms 1-4, CKB, CKV, and CKR, the outcome will be one in which the DM chooses a utility-maximizing strategy and nature chooses a state that has positive risk neutral probability given that strategy

• Fundamental theorem of games [2 or more DM’s vs. each other]: Given axioms 1-4, CKV, and CKR, the outcome will be one that has positive probability in a subjective correlated equilibrium based on common prior risk neutral probabilities

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Onward to Bayesian updating...• The paper goes on to assume that the state space S can be

partitioned as S = H I where I is a set of “informational” events that are expected to be resolved before an act is chosen and H is a set of “hypotheses” (consequential events) expected to be resolved only after an act has been chosen.

• Conditional preferences are defined in terms of contingent strategies that are pegged (only) to events in I.

• The DM is assumed to have no intrinsic interest in I-events, given the hypotheses, implying an observable likelihood.

• Optimal posterior decisions are made by updating risk neutral probabilities according to Bayes’ rule, then applying them to the evaluation of the original value gambles.