1 Decision Analysis Without Consequences, Bayesianism Without Probabilities Robert Nau Fuqua School of Business Duke University Presentation for RUD 2005 June 2005

2 Savage’s SEU theory: the foundation of “Bayesian” inference & decision analysis Key primitives: States of the world Consequences Key Axioms: Complete order (P1) Independence (P2) Values can be disentangled from beliefs (P3) Beliefs can be revealed by preferences (P4)

3 What is “Bayesianism”? Does it mean having beliefs that are represented by unique probabilities (i.e., does it require Savage’s P3 and P4)? Does it mean that individuals can observe each other’s “true” probabilities, or that they should (at least some times) agree on them? Or, does Bayesianism just mean having preferences that satisfy independence (P2), thus having an additive representation that supports conditionalization?

4 What are “consequences”? A natural consequence is the result of a feasible act of the decision maker and a possible state of the world Thus, acts logically precede consequences Savage does it the other way around: consequences are psychologically primitive “states of the person”, and acts are arbitrary (often counterfactual) mappings of states- of-the-world to states-of-the-person

5 What’s wrong with primitive consequences? States of the person are endogenous and unobservable—not a good foundation for descriptive and/or interactive decision theory Acts in which states of the world are mapped to counterfactual states of the person are purely imaginary—preferences can’t be validated by choices Preferences among acts yielding “constant” consequences still do not reveal “true” probabilities (e.g., utilities could have state-dependent scale factors)

6 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 (à la de Finetti)

7 Modeling framework Let A = {a 1, …, a I } denote a finite set of strategies (alternative courses of action) Let S = {s 1, …, s J } denote a finite set of states. Non-empty subsets of S are events. Let w = {w 1, …, w J } denote an observable wealth distribution over states (e.g., a side-gamble) An act is a strategy-wealth pair (a, w) An outcome is a triple (a, s, w) This is essentially the same state-preference framework used in Nau’s (Manag. Sci. 2003) model of generalized risk aversion, augmented by a finite set of strategies

8 s1s1 s2s2 … sJsJ (a, w) a, w 1 a, w 2 … a, w J (b, x) b, x 1 b, x 2 … b, x J (c, y) c, y 1 c, y 2 … c, y J …………… Acts States Each quantity of money received in a given state in conjuction with a given strategy is effectively treated as a distinct consequence

9 Properties of the decision maker’s preference relation  over acts Axiom 1: (Ordering)  is a weak order, and for every strategy the restricted preference relation over wealth distributions is a continuous weak order. Axiom 2: (Strict monotonicity) w* j > w j  (a, w* j w -j )  (a, w) for all a, j, w, w* j. (More money preferred to less, and no null states)

10 Axiom 3: (Strategic generalized triple cancellation) For every pair of strategies a and b, wealth distributions w, x, y, z, state j, and constants w j *, x j *: If (a, w)  (b, x) and (a, w* j w -j )  (b, x* j x -j ) and (a, w j y -j )  (b, x j,z -j ) then (a, w* j y -j )  (b, x* j z -j ). In words: if changing w j to w j * is no worse than changing x j to x j * when the background is a comparison of (a, w) vs. (b, x), then changing w j to w j * cannot be strictly worse than changing x j to x j * when the background comparison is (a, w j y -j ) vs. (b, x j z -j ).

11 Definition: Strategies a and b overlap if there exist w and x in the interior of  such that (a, w) ~ (b, x). There is a set of overlapping relationships that covers all the strategies if for any two strategies a and b, there exists a sequence {a, i 1, i 2, …, i k, b} such that a overlaps i 1, i 1 overlaps i 2, etc., up to i k overlaps b. Axiom 4: (Overlapping strategies) There is a set of overlapping relationships that covers all the strategies.

12 Main result: general additive model Theorem 1: Axioms 1-4 hold iff  is represented by a utility function U having the strategy- dependent additive-across-states form: U(a, w) =, in which {u aj } are strictly increasing, continuous utility functions for money that are unique up to joint transformations of the form  u aj +  j. Note: preferences among strategies without any side-gambles are represented by {  j u aj (0)}.

13 Example: 3 acts & 4 states s1s1 s2s2 s3s3 s4s4 (a, w)u a1 (w 1 ) + u a2 (w 2 ) + u a3 (w 3 ) + u a4 (w 4 ) (b, x)u b1 (x 1 ) + u b2 (x 2 ) + u b3 (x 3 ) + u b4 (x 4 ) (c, y)u c1 (y 1 ) + u c2 (y 2 ) + u c3 (y 3 ) + u c4 (y 4 )

14 Special cases of the general additive model 1.SEU with constant linear utility for $ u aj (w j ) = p j (v aj +w j ) 2.SEU with state-dependent linear utility for $ u aj (w j ) = p j (v aj +  j w j ) 3.SEU with state-indep. utility and prior stakes u aj (w j ) = p j u(v aj +w j +W j ) 4.General state-dependent SEU u aj (w j ) = p j v aj (w j )

15 What SEU preference parameters are “globally” observable (if entire preference order is known)? The DM’s “true” subjective probabilities for all events are observable only in special case 1. True probabilities are distorted by state-dependence and/or effects of prior stakes in cases 2 and 3 True probabilities are theoretically irrelevant in special case 4 (hopelessly entangled with utilities)! But… true probabilities are not needed for Bayesian decision analysis (or game theory)

16 Claim: decision analysis can be carried out in terms of “local” observations The entire preference order does not have to be observed—just a finite number of acceptable small gambles in conjunction with available acts Beliefs and values turn out to be only partly separable (i.e., “true” probabilities and utilities are not observed), but this doesn’t matter.

17 Local observables I: risk neutral probabilities The decision maker’s risk neutral (betting) probabilities are the normalized derivatives of the state-dependent utility functions for money: Note that they are, in general, act-dependent In the event that (a, w) is chosen, for whatever reason, z is an acceptable gamble if z · (a, w)  0

18 Local observables II: “value gambles” In the event act (a, w) is chosen when (b, x) is available, the DM should accept a gamble  (a, w; b, x) whose state- by-state increments of cardinal utility are proportional to the state-by-state differences in utility between (a, w) and (b, x), evaluated from the perspective of the local marginal utilities for money that apply under the chosen act (a, w). This implies acceptability of the “value gamble”  (a, w; b, x) determined by:

19 Decision analysis in terms of acceptable gambles Note: (a, w)  (a, w; b, x)  U(a, w)  U(b, x), Hence the direction of preference between any two acts can be determined from observations of acceptable gambles. This suffices to determine the optimal decision, even though “true” probabilities have not been observed (and may not exist…)

20 Prelude to Bayesian updating: conditional preference Let the decision maker be endowed with a conditional preference relation  E for every event E that reflects the updating of preferences with the receipt of new information. Axiom 4*: (Conditionally overlapping strategies) For any two events E and F, each consisting of 2 or more states, there is a set of overlapping relationships under  E that covers all the strategies and which is identical to a set of overlapping relationships under  F that covers all the strategies.

21 Conditional preference axioms Axiom 5: (Independence of irrelevant payoffs) (a, w)  E (b, x)  (a, w E w* ‑ E )  E (b, x E x* ‑ E ) for all w* ‑ E and x* ‑ E. Axiom 6: (Aggregation/sure thing principle) For disjoint events E and F, (a, w)  E (b, x) and (a, w)  F (b, x)  (a, w)  E  F (b, x)

22 Theorem 2: Given axioms *-5-6, for any event E consisting of 2 or more states, the conditional preference relation  E is represented by U E (a, w) = where{u ij } are the same state- and strategy- dependent utility functions that represent .

23 Hence, updating of preferences is accomplished simply by dropping ruled-out states from the utility summation This is consistent with the usual Bayesian updating procedure, even though probabilities are not separated from utilities.

24 Example: 3 acts & 4 states s1s1 s2s2 s3s3 s4s4 (a, w)u a1 (w 1 ) + u a2 (w 2 ) + u a3 (w 3 ) + u a4 (w 4 ) (b, x)u b1 (x 1 ) + u b2 (x 2 ) + u b3 (x 3 ) + u b4 (x 4 ) (c, y)u c1 (y 1 ) + u c2 (y 2 ) + u c3 (y 3 ) + u c4 (y 4 )

25 Preferences conditional on {1, 2} s1s1 s2s2 s3s3 s4s4 (a, w)u a1 (w 1 ) + u a2 (w 2 ) + u a3 (w 3 ) + u a4 (w 4 ) (b, x)u b1 (x 1 ) + u b2 (x 2 ) + u b3 (x 3 ) + u b4 (x 4 ) (c, y)u c1 (y 1 ) + u c2 (y 2 ) + u c3 (y 3 ) + u c4 (y 4 ) These states are dropped from the utility summation

26 State-dependent SEU interpretation s1s1 s2s2 s3s3 s4s4 (a, w)p 1 v a1 (w 1 ) +p 2 v a2 (w 2 ) + p 3 v a3 (w 3 ) +p 4 v a4 (w 4 ) (b, x)p 1 v b1 (x 1 ) + p 2 v b2 (x 2 ) + p 3 v b3 (x 3 ) + p 4 v b4 (x 4 ) (c, y)p 1 v c1 (y 1 ) + p 2 v c2 (y 2 ) + p 3 v c3 (y 3 ) + p 4 v c4 (y 4 )

27 Preferences conditional on {1, 2} s1s1 s2s2 s3s3 s4s4 (a, w)p 1 v a1 (w 1 ) +p 2 v a2 (w 2 ) + p 3 v a3 (w 3 ) +p 4 v a4 (w 4 ) (b, x)p 1 v b1 (x 1 ) + p 2 v b2 (x 2 ) + p 3 v b3 (x 3 ) + p 4 v b4 (x 4 ) (c, y)p 1 v c1 (y 1 ) + p 2 v c2 (y 2 ) + p 3 v c3 (y 3 ) + p 4 v c4 (y 4 ) These states are dropped from the utility summation, and probabilities are usually renormalized (but they needn’t be)

28 Special case: experimental information In a scientific experiment, the DM typically has no intrinsic in the experimental result given the hypothesis, in which case… The risk-neutral likelihood function is act- independent (reflective of “true” beliefs), while the risk-neutral prior is (as usual) contaminated by state-dependent utility, prior stakes, etc. Payoffs of value gambles depend only on the hypothesis, not the experimental result, and hence can be written in “reduced form”

29 Special case: experimental information Upon receipt of information, the risk neutral prior (for each available act) is updated via Bayes’ rule, by applying the common likelihood function. Preferences among acts are then updated by applying the risk neutral posteriors to the reduced-form value gambles. Hence, Bayesian methods do not actually require knowledge of the decision maker’s “true” prior probabilities!

30 Extensions/connections Risk aversion under the general additive model can be modeled via the methods in Nau 2003 (Management Science), though additivity rules out uncertainty aversion. The general additive preference model is also sufficent for “Bayesian” modeling of noncooperative games, where it leads to species of correlated equilibrium as the natural solution concepts (Nau and McCardle 1990 JET, Nau 1992 Management Science, Nau 1995 w.p.)