Several Bayesians: A review

Test (1993) Vol. 2, No. 1-2, pp. 1-32

Several Bayesians: A Review JOSEPH B. KADANE

Department of Statistics, Carnegie-Mellon University Pittsburgh, PA 15213-3890, U.S.A.

[Read before Spanish Statistical Society at a meeting organized by the University of Granada on Friday, February 12, 1993]

SUMMARY

This paper synthesizes a number of papers in which the author has participated that all concern models in which several decision-makers, each modeled as a Bayesian, appear. The areas covered include amalgamation of opinion, treating expert opinion as data, simultaneous-move games and sequential decision processes. The aim of the paper is to review the central ideas, and to explore how they relate to one-another.

Keywords: AMALGAMATION OF OPINION; EXPERT OPINION; GAME

THEORY (BAYESIAN); SEQUENTIAL DECISION PROCESSES.

1. INTRODUCTION

Standard Bayesian ideas suggest that a single decision maker, faced with uncertainty, should choose a decision d to maximize

f U(O,d)p(O)dO,

where p(O) is his opinion about the uncertain quantity 0 ~ f~, and U(O, d) is his utility for having made decision el when 0 obtains. The issue considered in this paper is how to extend this idea when there are several Bayesians involved in a problem.

The first literature to mention is ways of reducing a group to an individual. One method often considered is to amalgamate the decision

Received January 93; Revised April 93.

2 J.B. Kadane

makers in one of several ways so that the group becomes a new Bayesian with a group probability and utility. I review a bit of this literature, and explain why I am skeptical about it. A second idea treats the members of the group as providing data to a single decision maker to decide. While I think this is more successful, it also reduces the "groupness" of the decision-making. Both these ideas are taken up in Section 2.

A second literature concerns simultaneous decision problems, often modeled using game theory. A Bayesian alternative to game theory is reviewed in Section 3.

There is also Bayesian literature about sequential games, in which players take turns making moves. I review these in Section 4, and discuss their application to experimental design in section 5.

2. CAN SEVERAL BAYESIANS ACT AS ONE?

2.1. Amalgamation of Opinions and Utilities There is an extensive literature on amalgamation of opinion (Genest and Zidek (1986), French (1985)). The linear opinion pool is probably the most widely expressed idea: Let f l (0 ) , . . . , fk(0) be the opinions of k persons, then the amalgamated opinion is

k

i=1

where ~-~/k 1 wi = 1, wi >_ O. Note that g(0)is not necessarily anyone's opinion. DeGroot (1974) gives an interesting Markov model for how a linear opinion pool might develop. In DeGroot's model, there is a stochastic matrix P(k • k) representing how much each member of the group values the other's opinions. Then, having started with the vector (ft (0 ) , . . . , fk(O))' of opinions after one round, the new vector would be P( f l (0 ) , . . . , fk(O))', and after j rounds, PJ (fl (0 ) , . . . , fk(O))'. If P is irreducible and ergodic, then it has a steady state distribution, which can be taken to be the w's above. The division of the group into factions that put no weight on each other is modeled as non-communicating classes of states. Each of the classes come to a weighted consensus of its members but the classes are not brought to agreement by this process.

There are also various other pooling ideas such as logarithmic pools, etc. I will not pursue them further here.

Several Bayesians: A Review 3

This literature has not been extended to amalgamation of utilities, to the best of my knowledge, for reasons that I can only speculate about. Perhaps in the backs of our minds, while people are or ought to be free to have different goals (i.e., utilities), they ought not to have different opinions. I know of no particular reason why people should agree, and observe as an empirical fact that often people do not agree.

But my skepticism about this line of work does not end here. Sup- pose there are two Bayesians, Dick and Jane, whose utilities differ and whose probabilities differ. Suppose they seek to act jointly as a Bayesian would, and the only constraint they put on their compromise is that if both strongly prefer an option A to an option B, then their compromise cannot choose B over A. Under these conditions, Seidenfeld, Kadane and Schervish (1989) prove that the only compromises available are to do everything the way Dick wants to or the way Jane wants to: there are no intermediate Bayesian positions. Furthermore if the Pareto condition of agreement is strengthened slightly, so that if each party is indifferent or prefers A to B and one party strictly prefers A to B, then the compromise must prefer A to B, then even these two extreme "compromise" positions disappear. The Appendix to this paper gives some of the details on this result. The dissertation of Goodman (1988) pursues this theme when there are more than two decision-makers. If they agree in probability, convex combinations of utility are compromises. Conversely, if they agree in utility, convex combinations of probability (i.e., the linear opinion pool) are again compromises.

The implication of this result, to me, is that it will be very difficult to put the amalgamation literature on a proper decision-making footing. It is also difficult to apply. One unsuccessful attempt is reported in Genest and Kadane (1986).

2.2. How many sources of opinion can be influential to a single Bayesian

In a very important paper, Lindley, Tversky and Brown (1979) take up an issue of elicitation. They propose that the person whose opinions are being elicited be distinguished from the person doing the elicitation. The inferences being made are then all in the opinion of the latter, who will have opinions about both the phenomenon in question and about what the elicited-person's opinions might be. Statements by the elicited

4 J.B. Kadane

person are then regarded as speech-acts that are used as data. Once this perspective is adopted, an extension to several elicited persons is not difficult, nor is a change of emphasis from trying to discern the priors (i.e., elicitation) to the elicitator's opinion of the phenomenon itself. Thus what results is a model for how a Bayesian decision-maker might treat expert opinion. She has a joint opinion about 8, and also about the views of each of her experts, say 7ri (8), i = 1 , . . . , k. Hearing what they say (which is her data), she updates her opinions in this space. She may now emphasize elicitation if she wishes, by taking the marginal on the (7r1(0),..., 7rk(0))-space, or she may emphasize her own advance in knowledge about 8, by taking the marginal on 0.

While this model does make sense of a kind of amalgamation of opinion, it requires distributions on the space 7r1(8),..., 7rk(8) of opinions on 8. While some things can be done in such a space using Dirichlet Process Priors or Polya trees, doing so in a convincing manner still seems beyond what has been successfully modeled.

3. SEVERAL BAYESIANS, EACH MAKING DECISIONS

3.1. Several Bayesians Acting at the Same Time

Problems in which several actors make decisions simultaneously have played an important role in the history of Bayesian thought. These models, known as games, had a very influential impetus in the publication of the book of Von Neumann and Morgenstem (1944). They were pro- portents of the minimax approach to games, which in the simplest case of a two-person zero-sum game shows that if both sides randomize their strategies, a pair of such strategies exists in which neither side can im- prove its (expected) position by deviating. They also gave a version of expected-utility theory, but since their probabilities were "objective", it is not the modem, Bayesian version.

Such was the prestige of this book that it came to play a central role in the development of decision theory in statistics. Abraham Wald (1950) became one of the principal proponents of the adoption of minimax ideas into statistics, and L. J. Savage wrote in the early editions of his book (1954) that his purpose was to justify the minimax approach to statistics. In this he failed, as each of his rational models for decision-making came out to be Bayesian, not minimax.


Only gradually did he come to see that what he had derived was perhaps more interesting than what he had intended for it to'justify. Thus only gradually did Savage come to the view that if Bayesian statistics was unable to justify frequentist methods or minimax approaches, the fault might be in those methods and approaChes, rather than in Bayesian ideas. Bayesians spent (or misspent?) decades apologizing for the difference between their conclusions and classical ones.

The minimax approach to statistics has some inherent problems any- way. The idea is that the statistician is playing a simultaneous zero-sum game against Nature; and chooses to play a minimax, randomized strategy. The implication of this view is that whatever the statistician is trying to do, it is the purpose of Nature to thwart the statistician. Nonetheless, Nature is very benignly limited in her available moves: she cannot produce misleading data, nor choose the parameter values after seeing the data. In other contexts the belief that the word is specifically hostile toward a person could be regarded as a sign of paranoia, a mental ill- ness. By contrast, the Bayesian Nature is neutral toward the wishes of the statistician.

In Kadane and Larkey (1982) we returned to the theme of games, but from a Bayesian perspective. Even in the simplest case, a zero- sum two-person game, I cannot be sure that my opponent will play a minimax strategy. In fact, as a Bayesian, I have an opinion about what my opponent will do, and this is the source of my uncertainty in choosing what I should do. Having such an opinion, I choose my decision to maximize my expected utility. Only in the special case in which I am sure that my opponent will choose a minimax strategy will I be attracted to my minimax strategy, and even then it is in general only one member of a convex set of (mixed) strategies, each of which is as good as each of the others to me.

Thus in this Bayesian context the minimax strategy does not play a very favored or prominent role. The publication of this paper was not greeted warmly by all game-theorists (Harsanyi (1982), Aumann (1987)); a part of our response is given in Kadane and Larkey (1983). There is an alternate Bayesian line within game theory espoused by Harsanyi (1967) and Aumann (1987) that is based on the notion of a common prior distribution. The difficulty is that the space of the common prior involves the moves of all parties in the game. Thus Aumann invites

6 J.B. Kadane

you to bet against me about what I will choose to do, which appears to put you in the position of being a sure loser, contrary to Bayesian principles. More on this subject can be found in Kadane and Seidenfeld (1992).

One of the most intriguing games is the prisoner's dilemma, a two- person non-zero sum game. Two prisoners an separately held, and each is presented with the choice of whether to cooperate with the other or to defect by confessing to a crime. If one confesses and the other does not, the confessor will get a light sentence (1 year) and the other a heavy sentence (10 years). If bothconfess they will get moderate sentences (5 years) while if neither confesses they will get 2 years each on another charge. From the point of view of either prisoner, if the other confesses he is better off confessing (5 years rather than 10) and if the other prisoner does not confess he is better off confessing (1 year instead of 2). Hence each prisoner individually is better off confessing, but jointly they do better not to confess (2 years each instead of 5 years each). Thus confessing is the optimal move in a single play game. Now suppose the players face many such games against each other. By backwards induction, using classical game theory, they should confess every time. But a Bayesian approach to the repeated Prisoners Dilemma allows for cooperation between the prisoners if each believes strongly enough that a (risky) cooperative move early will be reciprocated later by the other player. Wilson (1986) explores this theme, and shows how to compute the optimal Bayesian strategies. See also Young and Smith (1991). Laskey (1985) discusses elicitation in this context.

The kind of thinking explored here using Bayesian strategies is applied in a very interesting book by Raiffa (1982) on negotiations. It does not explicitly use either game theory or Bayesian models, but its intellectual heritage is clear, at least to me.

3.2. Decision Structures with Several Decision Makers: Sequential Decisions

Sequential Decision processes are ubiquitous in real life, not only in classical games such as chess and bridge, but in many other contexts. They constitute an important area for Bayesian development.

In DeGroot and Kadane (1983) we undertook "an initial exploration into the question of what information it is necessary for Bayesians to specify in order to play a sequential game against each other". We did


so in the context of the simple three-move process: An object on the real line begins at so. Player 1 can move it to some other location Sl, then Player 2 moves it to s2, and finally Player 1 moves it to s3. Player 2's loss is proportional to the sum of the weighted squared distance of s3 from player 2's target y plus the squared distance player 2 moved the object, i.e., (s2 - Sl) 2. Similarly player l ' s loss is the sum of the weighted squared distance of s3 from player l ' s target x, plus a penalty equal to weighted squared distance that player 1 moved the object: (sl - s0) 2 plus (83 -- 82) 2.

When x and y are known to both players, simple calculus reveals the optimal Sl, s2 and s3 for each players. However one can also consider the situation in which each player knows his own target (i.e., player 1 knows x but not y, and player 2 knows y but not x). The nature of the solution is more interesting in this case. The last move, l 's , is very simple. Player 1 knows s2, where the object is, and x, his target. His move will be in the direction of x, and the amount of movement will be a function of the motion penalty compared to the penalty for the squared distance that s3 is from x. This move is the same whether Player 1 knows y or doesn't, because y is irrelevant.

The next stage backward, Player 2's move, is a bit more complicated. Player 2 knows the nature of what Player 1 's move will be, but doesn't know its magnitude unless he knows x, Player l ' s target. Hence Player 2's move depends on his prior for x, and, because of the simple nature of the loss functions assumed, depends on that prior only through its expectation. Finally, consider Player l ' s first move. His problem is that, whatever move he makes will not only move the object but will also be used by Player 2 as information about the location of Player l ' s target, x. Thus the expectation of x used by Player 2 will be conditional on Player l ' s first move, a matter that Player 1 must consider in deciding what first move to make. Hence even in this very simple three move decision problem, the player's beliefs about each other are essential ingredients to the solution. This game is reconsidered, with additions to represent "trembling hands" (Harsanyi and Selten (1988)), in Kadane and Seidenfeld (1992).

A line of work with a more applied flavor concerns the optimal use of peremptory challenges to choose jurors in the American legal system. In order to appreciate the problem, perhaps it would be helpful to review

8 J.B. Kadane

the institutional structure in which the problem occurs. The U. S. Bill of Rights (which are actually amendments to the Constitution) guarantee, among other rights, the right to be tried by a jury if one is accused of a crime. Many civil cases, which involve only money and not life or liberty, are also heard by juries. Most juries consist of twelve jurors, but some are as small as six.

The government, by a random selection process, chooses a panel of citizens who are potential jurors. By law (different, depending on in which state the case is heard), each side has a certain number of "peremptory challenges" it can use. These have the effect of excluding a particular potential juror from the jury. The challenges are called "peremptory" because the side using it need not explain why it wishes to exclude the particular juror in question, although recently the courts have held that racial criteria may not be used. Each side has the opportunity to pose certain questions to the potential juror before deciding whether to use one of its challenges to exclude that juror.

In a criminal case, a unanimous vote for guilt is required to convict. If a jury of size d is to be chosen, and side 1 (say the prosecution) re- gards the d jurors chosen as having, independently, probabilities, in the prosecution's view, P l l . . . , 1'1.1 of voting to convict, then the prosecution will wish to maximize I'I/J=l Pli. Similarly the defense will wish to

minimize H/J=I P2i. Supposing that jurors, characterized by (Pli, P'2i), come independently from a known joint cdf F(pl , P2), how should each side use its challenges, optimally, to get a jury it will most want to have? This question is taken up in Roth, Kadane and DeGroot (1977).

One of the assumptions made in the course of this work came to be called regularity, and amounts to the statement that each side prefers the situation in which it has one more challenge and its opponent one less. Several of our results depended on this assumption, but we were unable to prove it.

A later paper on this topic (DeGroot and Kadane, 1980) generalized the model by allowing each side to have a more general utility function and a more general view of the probability process. Under this model, we found a counterexample to regularity, but proved that it does hold if d - 1 or if the two sides have identical opinions (i.e., Pli = P2i for all i). Note that a counter example in a more general model does not settle the issue in the more specific model unless the counter example


satisfies the additional constraints imposed by the specific model, which it did not. Thus regularity in the Roth, Kadane, DeGroot (1977) model remained an open issue. DeGroot (1987) is an elegant summary of what was known about the problem at that time.

Subsequent work, done with two Carnegie Mellon undergraduates, Christopher Stone and Garrick Wallstrom, has shown that irregularity applies in the specific model as well. In addition, we show that if there are only two kinds of jurors, irregularity cannot obtain. Our counter example uses three kinds of jurors. This work is just now being written up.

An advantage of a Bayesian approach to sequential decision problems is that it offers a way of addressing skill in games. Skill is left out of the kind of game theory that has grown from Von Neumann and Morgenstern's (1944) work. Chess, for example, would be described as a completely deterministic sequential game, optimal strategies for which await only big and fast enough computers. While all this is true as far as it goes, it has little to do with chess as it is played. What constitutes skill in chess, how might it be developed, are there systematic ways to think about chess? Not having answers to these questions is perhaps why chess is still interesting to think about, but not having space in the description for understanding skill means that traditional game theory needs modification before it can be effective in addressing these issues. In Larkey, Kadane, Austin and Zamir (1993) we begin to address some of them, not for chess, but for a simplified version of poker. In a simula- tion, we find strategies that are pairwise-intransitive: A beats B, B beats C', and C' beats A. Such intransitivity means that great care is needed to define skill adequately.

4. A STATISTICAL APPLICATION IN EXPERIMENTAL DESIGN

Having shown above a plethora of models involving more than one Bayesian, a question remains of whether it is possible to make use of this for statistics. The best way to show that it can be useful is to use it for something.

In the spirit of last section, it is certainly reasonable to model using different Bayesians when there are several different kinds of decisions to be made. Traditional Bayesian experimental design has the same person, with the same prior, likelihood and loss function, play both the

10 J. B. Kadane

role of the designer, before the data collection, and the role of estimator, using the data to draw an inference. There is of course no reason why these two roles must be played by the same person, and in fact it is very reasonable to suppose that in practice often different people do these jobs.

As a start, Etzioni and Kadane (1993) suppose that Dan (the designer) and Edward (the estimator) share a normal likelihood with known precision, and a squared-error loss function. They each have a conjugate prior distribution on the mean of the normal likelihood, but with possibly different hyper-parameters. Suppose that Dan will choose a sample size (the very simplest experimental design question) and that Edward will estimate using expected squared error (and the expectation uses Edward's prior). Then Edward will choose his (Edward's) posterior mean, a precision-weighted average of Edward's prior mean and the sample mean. Suppose that Dan has to pay a constant cost per obser- vation plus the squared deviation of Edward's estimate from the truth. Then Dan's expected loss (with respect to Dan's opinion) involves how far apart Edward and Dan's prior means are, how sure each of them is, etc. Mathematically the equation for the optimal sample size is a cubic, replacing the quadratic obtained when Dan and Edward have the same prior.

This line of investigation is being extended by Madhumita Lodh in her dissertation work. I believe it is a fruitful line of development for a better un~terstanding of experimental design.

5. CONCLUSION

Clearly the general area of problems involving several Bayesians is one I have found intriguing for many years. The work reviewed here suggests the following:

1. With respect to amalgamation, if the work requires identical utilities, it is of limited usefulness. The Seidenfeld, Kadane and Schervish result casts doubt on whether an amalgamation formula can be successful.

2. The Lindley, Tversky and Brown approach, however, offers real hope for a personalistic amalgamation.

3. The Bayesian theory of simultaneous move games is a legitimate alternative to traditional game theoretic ideas.


4. The Bayesian approach to sequential decision processes is not simple, but has some useful aspects. The Bayesian theory of experimental design is open to generalization by distinguishing the designer from the experimenter.

My conclusion is that there is much left to be done, and everyone is invited to join in the fun.

6. APPENDIX: MORE DETAILS ON THE SKS IMPOSSIBILITY THEOREM

Let R be a finite set of rewards. AVon Neumann-Morgenstem lottery is a probability distribution P over R. Let 7r = (sl , . �9 �9 sn) be a partition of states. An (Anscombe-Aumann) horse lottery is a function from states to Von-Neumann-Morgenstern lotteries. Thus each prize in a horse lottery is a Von Neumann-Morgenstern lottery. Anscombe and Aumann (1963) make some standard assumptions on horse lotteries, also made here.

Suppose two Bayesian agents, 1 and 2, have utilities on Von Neu- mann-Morgenstern lotteries and probabilities on states. Suppose that these probabilities are not the same for some state, and that their utilities do not agree, i.e., P1 -r P2 and U1 ~ U2. Let <1 and <2 be their respective strict preference relations.

There are two possible Pareto conditions that might be imposed on their compromise ordering <:

Weak Pareto: If Epi,u i [A1] < Ep i,U i [A2] for i = 1, 2 then Ep, u [A1] < Ep, u[A2].

Strong Pareto: If Epi,u i [A1] < Ep i,U i [A2] for i = 1, 2 and at least one inequality is strict, then

Ep, u [A1] < Ep, u [A2].

Theorem (Seidenfeld, Kadane, and Schervish, 1989) Assume there is a pair of rewards (r . , r*) that the two agents

rank in the same order: r. <i r ' f o r i = 1, 2. Then

1. the set of Bayesian compromises satisfying the weak Pareto condition is { (P1, U1), (P2, U2) }, the two initial preferences.

2. the set of Bayesian compromises satisfying the strong Pareto condition is empty.

12 J. B. Kadane

R E F E R E N C E S

Anscombe, E J. and Aumann, R. J. (1963). A definition of subjective probability. Ann. Math. Statist. 34, 199-205.

Aumann, R. (1987). Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55, 1-18.

DeGroot, M. H. (1974). Reaching a consensus. J. Amer. Statist. Assoc. 69, 118-121. DeGroot, M. H. (1987). The use of peremptory challenges in jury selection. Contri-

butions to the Theory and Application of Statistics (A. Gelfand, ed.), New York: Academic Press, 243-271.

DeGroot, M. H. and Kadane, J. B. (1980). Optimal challenges for selection. Operations Research 28, 952-968.

Etzioni, R. and Kadane, J. B. (1993). Optimal experimental design for another's analysis. J. Amer. Statist. Assoc. (to appear).

French, S. (1985). Group consensus probability distributions: a critical survey. Bayesian Statistics 2 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. E M. Smith, eds.), Amsterdam: North-Holland, 183-201, (with discussion).

Genest, C. and Zidek J. (1986). Combining probability distributions: a critique and an annotated bibliography. Statist. Sci. 1, 114-148.

Genest, C. and Kadane, J. B. (1986). Combination of subjective opinion: an application and its relation to the general theory. Reliability and Quality Control (A. E Basu, ed.). Amsterdam: North-Holland, 141-155.

Goodman, J. H. (1988). Existence of Compromises in Simple Group Decisions. Ph.D. Thesis, Carnegie Mellon University.

Harsanyi, J. (1967). Games with incomplete information played by 'Bayesian' players. Management Science 14, 159-182; 14, 320-334; 14, 486-502.

Harsanyi, J. (1982). Subjective probability and the theory of games. Management Science 28, 120-125. Comments on Kadane and Larkey's paper with reply and rejoinder.

Harsanyi, J. and Selten, R. (1988). A General Theory of Equilibrium Selection in Games. Cambridge: University Press.

Kadane, J. B. and Larkey, P. (1982). Subjective probability and the theory of games. Management Science 28, 113-120.

Kadane, J. B. and Larkey, P. (1983). The confusion of is and ought in game theoretic contexts. Management Science 29, 1365--1379.

Kadane, J. B. and Seidenfeld, T. (1992). Equilibrium, common knowledge and optimal sequential decisions. Knowledge, Belief, and Strategic Interaction (C. Bicclaieri and M. L. Dalla Chiara, eds.), Cambridge: University Press, 27-45.

Larkey, P., Kadane, J. B., Austin, R. and Zamir, S. (1993). Skill in games. Tech. Rep. Heinz School, Carnegie Mellon University.

Laskey, K. B. (1985). Bayesian Models of Strategic Interaction. Ph.D. Thesis, Carnegie Mellon University.


Lindley, D. V., Tversky, A. and Brown, R. (1979). On the reconciliation of probability assessments. J. Roy. Statist. Soc. A 142, 146-180, (with discussion).

Raiffa, H. (1982). The Art and Science of Negotiations. Cambridge: University Press. Roth, A., Kadane, J. B. and DeGroot, M. (1977). Optimal peremptory challenges in

trial by Juries: a bilateral sequential process. Operations Research 25, 901-919.

Savage, L. J. (1954). Foundations of Statistics. Chichester: Wiley.

Seidenfeld, T., Kadane, J. B. and Schervish, M. (1989). On the shared preferences of two Bayesian decision-makers. Journal of Philosophy 5, 225-244. Reprinted in The Philosopher's Annual 12, 243-262.

Von Neuman, N. J. and Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton: University Press.

Wald, A. (1950). Statistical Decision Functions. Chichester: Wiley.

Wilson, J. (1986). Subjective probability and the prisoner's dilemma. Management Science 32, 45-55.

Young, S. C. and Smith, J. Q. (1991). Deriving and analyzing optimal strategies in Bayesian models of games. Management Science 37, 559-571.

DISSCUSSION

JAVIER GIRON (Universidad de Mdlaga, Spain )

Let me congratulate Professor Kadane by this most interesting paper on the difficult subject of decision and statistical problems involving several Bayesian decision makers. I must say that, whilst reading the paper, I was impressed by the larger number of contributions to this and related topics made by the author. I have found specially interesting the section dealing with the Bayesian approach to games and the idea of incorporating the concept of skill in Game Theory, and the novel look he presents to the old problem of experimental designs, distinguishing between the roles of the designer and the estimator.

The topics covered by the paper are basically the following:

i) Amalgamation of opinions ii) Treating expert opinion as data

iii) Bayesian games procedures iv) Sequential decision procedures (including new research on the con-

cept of skill) v) Applications to experimental designs.

14 J. B. Kadane

However, I will concentrate my contribution on the aspects of the paper I am more familiar with; namely, amalgamation of opinions and of utility functions.

Although Professor Kadane is skeptical about the amalgamation of opinions, there is still some research going on this subject as shown by some recent papers (see, e.g., DeGroot and Mortera (1991) and Gilardoni and Clayton (1993)).

Methods of amalgamation and Bayesian updating. The linear opinion pool (LOP) and the logarithmic pool (GM) are the most commonly used procedures for amalgamating opinions. In this section I will comment briefly on the relationship between these methods of amalgamation and the updating of opinions via Bayes theorem when new information (and the same for all, i.e., the same likelihood functions is shared by all individual) is available. The next diagram shows the two possible ways of doing this.

l Opinions ]

Combination or amalgamation of opinions

/

Bayesian individual updating of opinions

Bayesian updating of the combined distribution

Combination or amalgamation of posterior opinions

Raiffa (1968), Chapter 8, Sections 10 and 13, recommends using the LOP before applying Bayesian updating, i.e., to follow the upper path in the diagram.

The problem of seeing if both ways give the same final answer was termed external Bayesianity by Madansky (1978). He showed that the GM procedure is externally Bayesian while the LOP is not. The problem is that, in the LOP, the weights should also change with data, as follows from Bayes theorem.


In fact, using the same notation as that in the paper, if f l ( 0 ) , . . . , fn (0) are the opinions of k persons and the (prior) weights are wi >__ 0

k with ~-~i=1 wi = 1, then the LOP is

k

:Lop(O) =

i=1

If x is the result of an experiment and L(O; x) denotes the common likelihood function, then the posterior of the amalgamated prior is

k

fLop(Olx ) = ~ w~(x)fi(OIx), i=1

where fi (0Ix) are the posteriors of the k individuals and the (posterior) weights w~(x) are given by

~x f L(O;x)fi(O)dO.

In this way updating both the priors and the weights we obtain Bayesian externality for the LOP.

Bayesian updating (fi(O),...,fn(O)) ' (fi(alx),...,fn(Olx))

OoOroot's I iOeOroot's method method

Bayesian updating f*(o) f*(OIx) = f(OIx)*

An interesting model that justifies, and develops a method for the LOP is DeGroot's method for reaching a consensus which Kadane de- scribes briefly. In the paper by Caro, Domfnguez and Gir6n (1984) it is shown that DeGroot's method is also compatible with Bayesian updating, as shown in the above diagram, where f* (0) denotes the consensus

16 J. B. Kadane

distribution before observing the data x, f*(OIx) the posterior of this consensus distribution and, finally f(OIx)* denotes the consensus obtained by applying DeGroot's method to the posterior distribution of the k persons.

The key idea in proving this result is that, after the Bayesian updating at each round, the initial transition matrix P also changes with data and with the round if Bayesian updating is applied before DeGroot's method.

Amalgamation of utilities. The combination of utility functions is, in a sense, more complex than that of amalgamating opinions due to the fact that they do not change with data so that a consensus is generally impossible. As Kadane explicitly suggests, people should not agree. We also believe that risk attitudes are proper to the individual and they might change with time but not with data.

Yet, from a formal mathematical viewpoint probability and utility are dual concepts. In fact, the subjective expected utility

/ E[u; P] = udP can be regarded as a bilinear functional where u ranges in a space of functions and P in a space of measures, which sometimes are dual spaces in the sense of Functional Analysis.

Based on this idea, a method for amalgamating utility functions, analogous to the LOP would be as follows:

Let :P be a set of risk prospects, described, for example, by a subset of the set of distribution functions on some, say, finite interval I = [a, b], Jr(I).

Let ui, i = 1 , . . . , k be the normalized, i.e., u(a) = O, u(b) = 1, utility functions of k decisions makers.

Each ui generates a weak linear order __.i in Jr(I)

F ~i G z. ;. f i u i d F > f iu idG.

The intersection of these preorders produce a weak partial order

F'z- , ' f lU idF >_ foraU i = l , . . . , k ;

and the maximals in P with respect to ~ give the Pareto (non-dominated) solutions for the group.


The linear utility pool (LUP) can be defined by the utility function k k ~-,i=1 wiui, where the weights wi > 0 and ~,i=1 wi = 1. The new weak

order generated by the pooled utility function ___u is compatible with the partial weak order ~ as u belongs to the convex cone spanned by the ui.

One advantage of the LUP is that if all decision makers have, for example, risk aversion or have DARA utility functions, then the pooled utility function also presents risk aversion or belongs to the class of DARA functions, respectively.

It would be interesting to characterize axiomatically the LUP, in a similar way to the known characterizations of the LOP, using the duality between probability and utility suggested above in this section.

DANIEL PE/qA (Universidad Carlos III de Madrid, Spain ) This is a very stimulating paper on an important topic and the author should be congratulated for his review of this challenging research field.

I will concentrate my comment on the amalgamation problem. The advantages of the linear opinion pool are (Genest and Zidek (1986), French (1985)) the marginalisation property, (the same result is obtained whether (1) their opinions are first combined and then a marginal distribution is taken; (2) the individual marginals are obtained and then they are combined); the Zero Probability property (if (Pi (A) = 0, then Pa(A) = 0, when / indexes individuals and g is for the group), and its simplicity. I will add that the linear opinion pool has also the advantage of providing a mean for the group distribution that is a precision-weighted average of the individual means. This result is in agreement with the standard solution for combining independent sources of information in classical statistics: it is well known that if we have k independent un- biased estimators 0i of a parameter tO with precisions Pi, the efficient estimator is

~g : E Pi ~. (1) Epi

the precision-weighted average of these independent estimators. Also, in standard Bayesian inference for the mean of a normal distribution with a conjugate normal prior, the mean of the posterior has this interpretation. De Groot and Mortera (1991) have shown some of the advantages of linear opinion pools in some particular problems.

However, the linear opinion pool may have all the drawbacks of the naive combination of non homogeneous data. For instance, in the

18 J. B. Kadane

simplest case of two experts with normal priors N(pi, ~ ) , i = 1, 2, 2 (i = 1, 2) the group distribution will be a bimodal when [#1 -/z21 > O" i

mixture of normals. This result could be acceptable if the differences between the two members are due to their difference of opinion about some event that may or may not occur, and whose presence justifies the extreme values for the means ~1 and #2. But if this is not the case, it may well be that the group opinion will be better modelled by a normal distribution with parameters that are obtained as a weighted average of the individual ones.

The combination of non homogeneous data can also lead to Simp- son's Paradox. Suppose, for instance, that the probabilities that 3 experts assign to an event A are .70, .20, .50, and to another unrelated event B are .75, .25, .55. Let us assume that the relative credibility given to these experts are .4, .1 and .5 for their evaluation of the likelihood of A, and �9 1, .4 and .5, for the evaluation of B. Then, the linear opinion pool leads to a group probability of .55 for A and .45 for B. Therefore, although the three members of the group agree that event B is more likely than event A, their joint opinion built by the linear opinion pool is opposite to their individual opinions.

The Bayesian updating approach advocated by Lindley, Tversky and Brown (1979), French (1981), and Lindley (1985), among others, seems to offer a straightforward way out to this problem: suppose your prior probability distribution for some unknown quantity is 7r(0) and you "observe" that a given expert has a probability distribution pi(O) over 0. You can consider this piece of information as data, and update your current beliefs by

7r(Olpi, ~r) = cp(pi[O, 7r)Tr(O) (2)

where the likelihood p(PilO, 7r) express your opinion about the subjects. Lindley (1985) has shown some examples of how this likelihood can be built when 0 is discrete, but in the general case the problem is very difficult to solve.

As an example suppose that we are interested in the mean of a Normal distribution O, the decision maker has a normal prior over 7r(0) ,-, N(#0, or02), and the experts have normal distributions Pi (0) N N ( # i , 0"/2), i = 1 , . . . , k. Then, Lindley's approach requires the definition of a likelihood over each of these probability distribution and this is clearly not an easy task to do.


I suggest the application of the Bayesian paradigm in this problem in a more straightforward way. Suppose that instead of thinking about likelihoods P(Pi 10, 7r) over probability distributions the decision maker defines the precision he/she assigns to the experts' distributions, that is, he/she defines each expert reliability by some coefficients a12,..., a 2. Let us assume that the experts have, a priori, independent opinions. Then, the posterior distribution of the decision maker after observing Pl (0) with reliability a l 2 is

( ' a12a12#l ; 0 0 2 + a l 2) a ~ -2 7r(0lpl) "-~ % z _q_ al O. 0 q_

and the posterior opinion when all the experts distribution are "ob- served", will be

r(0[pl, ,Pk) N ' " a:z 2 �9 . . ,~ ~---fL~_2 7ri~ kSa? o

with a0 = a0. In summary, this procedure can be summarized as follows: (1) use the decision maker distribution, rr (0) as prior distribution for the problem; (2) use the experts' opinions Pl (0) . . . , Pk (0) as data, and define the equivalent information content included on each distribution with respect to the prior (that is the coefficients a~ -2, %2 = ao2); (3) update information sequentially using Bayes' theorem.

The advantages of this approach are: (1) it can be easily applied in standard problems; (2) it implies that the decision maker's information increases with experts' information; (3) the factor corrections (aj lai) 2 provide a way to calibrate experts. Of course, some extension of this approach can be made to (1) incorporate dependency among experts; (2) include restrictions on the factor corrections; (3) provide practical ways to assess the precisions ai.

I would like to comment briefly on the amalgamation of utilities�9 If the group behaves in a coherent way it will choose a decision d to maximize

l ug(O, d)pg(O)dO (3)

where ug and pg are the utility function and probability distribution for the group. On the other hand, each member will choose d to maximize

f ui(O,d)pi(O)dO (4)

20 J. B. Kadane

where ui and Pi correspond to the individuals. A Pareto optimum for the group implies jthat the decision must be taken to maximize

~.~ wi Jfl ui(O, d)pi(O)dO, wi >_ 0, ~wi = 1. (5)

A key question is to know whether or not (3) is compatible with (5). The difficulties with this approach were recognized by Wilson (1968) and Raiffa (1968), who studied some of the conditions for this compatibility to hold and some of the paradoxes that may occur. Again, as a result of mixing up linearly non homogeneous pieces of information Simpson's Paradox can easily happen.

Finally, let me express my whole agreement with the Bayesian approach for analyzing games advocated in the paper, following Kadane and Larkey (1982, 1983) and Raiffa (1982). As indicated by Rapa- port and Chamman (1965) in game theory "strategically rationalisable courses of action are frequently intuitively unacceptable and vice-versa". The prisoner's dilemma is an interesting example of it, and the advantages of Bayesian ideas have recently been shown by Young and Smith (1991), providing a graphical procedure to identify the form of an optimal solution.

The following contributions were later received in writting.

PETER FISHBURN (AT & T Bell Laboratories, U.S.A. ) It is pleasure to thank Professor Kadane for his review of highlights of his research on multiperson Bayesian decision theory. I have followed this work for many years and admire his clear explanation of its topics. Although an overview can only hint at the fascinating mathematical and interpretational issues involved, Professor Kadane gives a good account of the terrain and its challenging problems. I would add a few remarks on aggregation, game theory, and the research scene.

My main comments are motivated by the striking impossibility theorem of Seidenfeld, Kadane and Schervish in the setting of Savage's formulation of decision under uncertainty. Their theorem says that if two Bayesians' subjective expected utility representations for preferences between acts differ in their subjective probability and utility functions, and if a third preference relation has act A preferred to act B whenever one of the two prefers A to B and the other also prefers A to B or is indifferent between them (strongly Paretian aggregation), then the third


relation can not be represented by a subjective expected utility model. This lies in contrast to Harsanyi's (1955) linear aggregation theorem for von Neumann-Morgenstern utilities, and to the Pareto aggregation theo- rems for decision under uncertainty which say that convex combinations of subjective probabilities (respectively, convex combinations of utilities) are Bayesian admissible when two or more persons have similar utilities (respectively, identical subjective probabilities).

Professor Kadane mentions a way around the impossibility im- passe which treats the subjective probabilities and utilities of disagree- ing Bayesians as input data for an opinion amalgamator, who might be thought of as the primary decision maker. I refer to this person (or process) as a Bayesian aggregator when his or her preferences satisfy a subjective expected utility model. Impossibility tells us that a Bayes- ian aggregator must violate the strong Pareto condition for the (other) opinions being aggregated unless they exhibit unrealistic unanimity in probabilities or utilities.

An interesting alternative to a Bayesian aggregator is a Paretian aggregator, which I define as a person or process that always declares A preferred to B when every person in the aggregation pool prefers A to B or is indifferent between them, and at least one strictly prefers A to B. A Paretian aggregator can always have transitive preferences (assuming weak orders in the pool), but must be non-Bayesian when unanimity of probabilities or utilities fails.

Paretian aggregators exist in abundance, and if we are willing to give up some properties of a Bayesian aggregator then certain Paretian aggregators may be attractive. I shall not speculate on what they might look like, but note that two lines of inquiry could facilitate their examination. The first line involves generalizations of the Bayesian subjective expected utility model that are described, for example, in Machina (1987), Fishbum (1988), Karni and Schmeidler (1991) and Camerer and Weber (1992). The second line takes the approach of social choice and voting theory. Kelly (1991) offers a recent bibliography of this area, and Fish- burn (1987) reviews developments that are more narrowly focused on aggregation theory's impossibility results.

Turning to Bayesian perspectives in game theory, I vividly remem- ber the 1982 appearance in Management Science of the Kadane-Larkey paper proposing a straightforward Bayesian solution concept for non-

22 J.B. Kadane

cooperative games as an alternative to minimax-type solutions. I found their approach refreshing as well as eminently sensible, and was sur- prised by the controversy it generated. In retrospect, it seems that their paper has been good for game theory, and it is encouraging to see that Professor Kadane remains active in this area.

His invitation to others to engage in multiperson Bayesian research is most welcome. In view of the vast quantity of research on a single Bayesian decision maker, and the social nature of many decision processes, I find it surprising that more has not been done on the multiperson Bayesian level. We are indebted to Professor Kadane and his collaborators for their pioneering work on that level and for directing others toward what will surely be a prime research area in the future of decision theory.

SIMON FRENCH (University of Leeds, U. K. )

Professor Kadane's paper Several Bayesians: a review reflects on many issues in the Bayesian modelling of group interactions. He offers many pertinent comments, and I have little quarrel with what he says. I enjoyed his paper greatly and I congratulate him on it.

In particular, I share his concerns about some of the directions taken in game theory over the years (French, 1986). Game theory, it seems to me, fails as a descriptive theory: people may be pessimistic, but they are not consistently pessimistic! It also fails as a normative theory because, as Professor Kadane argues, minimax is not normatively appealing. Thus game theory it not very useful in developing prescriptive methods (Bell et al., 1988) which combine descriptive and normative models to help decision makers. Strictly, I am not correct here, and the cause of my inaccuracy is interesting. Both Nigel Howard's use of metagames and Peter Bennett's use of hypergames show that game theory can led to methods which do help decision makers (see, e.g., Rosenhead 1988). But, and here is the rub for game theorists, Howard and Bennett use game theoretic concepts much more to help in problem formulation, in identifying potential strategies and generally in structuring issues. There is very little reliance on formal preference and belief modelling, such as would underlie the use of minimax. If game theory is to contribute further to the support of decision makers, I believe that it will need to develop along the lines suggested by Kadane and Larkey (1982, 1983).


Having agreed with what Professor Kadane has said, let me move on to point to what, I believe, are some omissions. Firstly, his paper avoids discussion of the concept of calibration. To be fair, he has already contributed to that discussion elsewhere (e.g., Kadane and Lichtenstein, 1982). In the modelling of the beliefs of several Bayesians, the difficulty with calibration is that, in essence, one Bayesian's probabilities are calibrated for him or her, but not for another Bayesian; and vice versa. This means that it is possible for a single decision maker to assimilate the opinions of several experts, as in the Expert Problem (French, 1985). But it is not possible, except in particular circumstances, for a group of Bayesians to come to share a common probability distribution.

Professor Kadane points to the intractability of Bayesian solutions to the Expert Problem. But the problems are important practically and there are many efforts underway to provide useful solutions: see ESS- RDA (1990) and Cooke (1991) for recent reviews. Most of this work is non-Bayesian being based upon weighted opinion pools: that should give Bayesians some cause for concern. Perhaps some hope is offered by the Bayesian methods developed by Michael Wiper and myself (French and Wiper, 1991), particularly when combined with some of the recent developments in the use of Gibbs samplers and other Monte Carlo methods.

The focus in Professor Kadane's paper is on combining the beliefs of several Bayesians through the use of models. One should not ignore the progress that is being made in working directly with groups of Bayesians and eliciting an agreed set of beliefs: see, e.g., Kaplan (1988). Indeed, moving from problems of combining beliefs to problems of agreeing on a course of action, i.e., group decision problems, there is much work on using (the spirit of) the Bayesian approach within decision conferences: see, e.g., French (1992) for a review.

To close: may I again thank Professor Kadane for his stimulating paper? I hope it encourages more to 'join in the fun'. There are plenty of important practical problems to solve.

D. V. LINDLEY (Somerset, U.K. ) In the case of a single decision-maker, there is a normative theory, stated concisely by Kadane at the beginning of his paper, which is that the decision procedure must be Bayesian. The important feature to notice, as far as this paper is concerned, is that there is no normative theory when

94 J.B. Kadane

there is more than one decision-maker. Indeed, such results as we have, like Arrow's, suggest that there cannot be a theory. This is a puzzling state of affairs, and is especially worrying from a practical viewpoint, when there are so many areas of conflict between decision-makers.

One way in which we often avoid the difficulty in practice, is to elect one of them as a principal who then decides, taking into account the others' views, as described in Section 2.2. This reduces the problem to one with a single decision-maker and the normative theory applies. The manner of the election remains unsolved. Incidentally, Kadane is perhaps too pessimistic here in noticing how the leader takes account of the other views. Several quite complicated situations have been successfully tackled: for example, West (1992).

Why is the problem of multiple decisions-makers so difficult, even in the simple form of the prisoners' dilemma? It is because one of them, A say, can take account of B's ideas by Bayes, but when A thinks about B, the realization comes that B is thinking about A. But in B thinking about A, thought will also go into what A thinks about B. The cycle continues indefinitely and remains unresolved. Notice that the prisoners' dilemma remains a problem even when they can communicate, for if they agree neither to confess, each has still an advantage in changing.

Kadane makes the point that there has been little work on the combination of utilities. I would go further and say there has been little work on utility by statisticians. The reason for this is perhaps historical. Statisticians used to regard themselves as the gatherers of data and would let the data speak for itself. Later they began to help in the speech by developing ways of analysing data. More recently they have, in the Bayesian paradigm, discovered how to use the data to decide. They have still to make the next step and use their new expertise.

Finally, on the question of experimental deign, I think that Kadane's emphasis is correct and that, as a result, we have a new tool for this field. Lindley and Singpurwalla (1991) provide a practical illustration.

GIOVANNI PARMIGIANI (Duke University, U.S.A. ) This article summarizes a remarkable array of ground-breaking contributions to Bayesian multi-decision-maker problems. I would like to use the opportunity to comment on it by adding one small, and mainly dec- orational, historical footnote concerning the Kadane and Larkey (1982) approach to zero-sum two-person games.


I think it can be argued that the Kadane and Larkey (1982) proposal, so controversial among game theorists, would have found a full endorse- ment from Bruno de Finetti. In fact, in a rather obscure reference, de Finetti expressed similar views. The brief quotation that follows is an excerpt from an encyclopedia item entitled "Decisione" (in Italian). de Finetti 's goal is introducing the basic concepts of decision-making under uncertainty to an audience of non experts. After presenting an example of a two-person zero-sum game, with a six by six payoff tables, he writes:

� 9 from the point of view of individual A, we must choose one of six columns, each of which includes six values...; the actual gain or loss depends on the choice of the row, which is made by the opponent B (without knowing which is column chosen by A). What probability should A assign to B's choice of the row? (B will evidently try to similarly assign probabilities to Ns choice). This is a doubly psychological problem, as each player is forced to ponder Dante's Cred"t'o ch'ei credette ch'io credesse [Inferno, XIII, 25], thinking: "What evaluations would I give thinking of those that my opponent will give trying to imagine mine?"

... Game theory, rather than looking at the problems in these terms, that are its true terms, prefers to resort to adhockeries, . . . and consider different questions such as "How can I avoid (with certainty) results that are too unfavorable?"

It appears that de Finetti would consider it natural to assign a prior to the opponent 's strategy in solving the game. Also, he seems to be viewing the full rationality assumption as a way of circumventing the difficulty of assigning such prior.

This remark made in passing is the only place in which, to my knowledge, de Finetti commented on this issue. The article quoted does not develop the idea any further. Yet, there is enough to conclude that de Finetti would have enjoyed the Kadane and Larkey (1982) paper.

ROBERT L. WINKLER (Duke University, U.S.A. ) Inferential or decision-making problems involving multiple persons in one way or another are especially interesting, intriguing, challenging, and often frustrating. Whether we are talking about several persons who must jointly make a decision, an analyst who must combine the judgments from a number of experts, a competitive situation with several players, a market with numerous buyers and sellers of a good, a single decision maker whose decision will affect a wide range of people

26 J. B. Kadane

(perhaps the entire population of a country, for example), or yet some other scenario, multiple-person problems can be very tricky to study and model. As soon as a second person enters the picture, the model has to represent a new layer of complexity, with thorny issues such as interpersonal comparisons of preferences often standing in the way of a "clean," simple model.

Multiple-person problems are also the rule rather than the exception in practice. In today's society, decisions in both the public and private sectors are typically made by groups or at least involve widespread con- sultation, information is usually available from a number of experts and other sources, and the trend is certainly toward market-based economies as of this writing. As a result, multiple-person problems have been studied from widely divergent viewpoints, with the fields involved including statistics, management, operations research, economics, psychology, so- ciology, engineering, mathematics, and public policy.

Although statistics is listed first in the preceding list and some no- table contributions to the modeling of multiple-person problems have come from statisticians, I would say that statistics has a lower profile in this regard than might be expected. Kadane is one of a surprisingly small number of statisticians who have delved deeply into a variety of issues involving models of multiple-person problems, and he has done so in an interdisciplinary fashion, making valuable contributions not just to statistics but to some of the other fields noted above. That is why it is particularly pleasing to read his personal review of this area. Kadane discusses many meaty issues in a relatively short paper and whets the reader's appetite for more.

I share Kadane's interest in multiple-person problems of many different types and will try to briefly add my own personal thoughts to Kadane's excellent review. In so doing, I take up the spirit of Kadane and emphasize work in which I have participated, adding a few historical notes as well. The ordering of topics more or less follows that of Kadane.

The amalgamation of opinions, sometimes called consensus or expert resolution or the combination of probabilities or forecasts, has a long history. The notion of a decision maker obtaining probabilistic judgments from a number of experts and combining them in a Bayes- ian spirit is discussed in Winkler (1968), where several combining ap-


proaches are suggested. The first full-blown formal Bayesian development is presented in Morris (1971, 1974), which predates Lindley, Tversky, and Brown (1979) by several years. A recent review that em- phasizes the combination of forecasts is given in Clemen (1989). Both theoretical and empirical work indicate that the typically-encountered high dependence among experts creates difficulties for amalgamation. Clemen and Winkler (1993) develop a Bayesian framework for combining forecasts featuring a highly flexible environment for modeling the likelihood function and dealing with interrelationships among the experts. Work is currently proceeding on applications of this approach to the combination of probabilities.

The amalgamation of utilities also has a long history, albeit not without some controversy. Harsanyi (1955) presents conditions under which a group cardinal utility function can be expressed as a linear combination of the utility functions of the individuals comprising the group. Keeney (1976) provides alternate conditions leading to a group utility function. The amalgamation of utilities to form a group utility function is modeled in Eliashberg and Winkler (1981) with each individual's utilities assessed not just for his or her own payoffs (or more general consequences), but for the entire vector of payoffs received by the group members. This approach enables each individual to include any preferences for features such as equity among the group members, and the resulting group decisions and allocation of group consequences within the group will be sensitive to such preferences. Much of the literature relating to utility amalgamation has appeared under the broad heading of social choice theory (see, e.g., Fishburn, 1973), and equity seems particularly salient in social decision making. Keeney and Winkler (1985) distinguish between ex ante and ex post risk equity in decisions that include possible loss of life among the possible consequences and treat both types of equity in avon Neumann-Morgenstern utility model developed to evaluate public risks.

The controversy surrounding the amalgamation of utilities has focused primarily on the very basic issue of the validity of interpersonal comparisons of preferences. Moreover, the underlying behavioral ax- ioms supporting utility theory seem less clear-cut and less compelling to some when the context is that of a group utility function as opposed to an individual utility function. More generally, potential paradoxes and

28 J. B. Kadane

problems arise when members of the group have different probabilities and different utilities, as noted by Kadane (see also Raiffa, 1968, Chap- ter 8, for some interesting examples and results). This is fertile ground for further work, including the possibility of negotiation among group members to expand the set of altematives in an attempt to find "win- win" solutions (Raiffa, 1982). Keeney's (1992) value-focused thinking, which is relevant for both individual and group decision making, is in the same spirit.

Of course, the "several Bayesians" could be making individual decisions in a competitive situation instead of trying to make a group decision. This takes us into Bayesian game theory, which is a fascinating but difficult field. In a game involving two Bayesians, each revises his or her probabilities based on any statement or action by the other. Con- cepts such as the value of information become much more complicated than in games against nature; for an example with a sequential game, see Ponssard (1976). In some cases the value of information to a player can be negative because the other player knows the first player has the information (but not what the information says) and modifies strategies as a result. Different types of information, such as secret information (not seen or known about by competitors), private information (known about but not seen by competitors), and public information (seen by all) might be considered. Different utility functions can influence the outcome of a game (Eliashberg and Winkler, 1978). Moreover, despite the typical assumption in game theory that the probabilities and utilities of the players are common knowledge, the typical situation in practice is that each player is uncertain about the probabilities and utilities of the other players. In this context, the fact that a group decision-making problem can be modeled as a game with the players being the members of the group underscores the potential difficulties in models of group decision making. Kadane and Larkey (1982) make a valuable contribution in viewing a zero-sum game from a Bayesian perspective without the common knowledge assumption, and Nan and McCardle (1990) invoke Bayesian foundations by showing that Aumann's notion of a correlated equilibrium can be developed from a de Finetti-like perspective involving the avoidance of Dutch books.

Markets for financial instruments and other commodities can be thought of in terms of Bayesian games, of course. The role of market


makers, who might have access to more information than most investors, is studied in Jaffe and Winkler (1976) and Conroy and Winkler (1981, 1986). A Bayesian model of trading against a market maker shows how an investor might revise probabilities for future stock prices based on the buy and sell prices offered by the market maker. If the market maker is viewed as better informed than the investor, the investor should trade only if the initial probabilities regarding future prices are quite divergent.

In conclusion, I eagerly await many further contributions from Ka- dane to the modeling of multiple-person problems, and I heartily endorse his closing comment: "there is much left to be done, and everyone is invited to join in the fun"

REPLY TO THE DISSCUSSION

I thank each of the discussants for careful and interesting consideration of the issues raised in my paper. I review below their comments by topic, in the same order as the paper.

The amalgamation of probabilities is commented on extensively by both Gir6n and Pe~a. Gir6n reports on the literature concerning whether particular amalgamation methods and Bayesian updating com- mute. Amalgamation of utilities is commented on by Gir6n (showing an interesting duality) and Pefia. Both French and Winkler discuss the practical need for amalgamation methods, and the literature on it.

With respect to the impossibility theorem, I did not mean to imply that I value theory over practice. Good theoretical understanding can aid practice; and practical experience can lead to deeper and better theoretical questions. However, in this case I am not sure that the implications of the theorem have been entirely worked out. It is desirable to have a "decision theory" for a group that mirrors decision theory for individuals. Accepting the Pareto condition, the impossibility theorem says that this goal cannot be reached with Bayesian Decision Theory. It seems to me that this leads to three alternatives: (i) abandon the idea that group rationality should match individual rationality (Fishburn's interesting idea of Paretian aggregators is along these lines), (ii) abandon Bayesian Decision Theory as a normative standard for individuals (there is a huge literature exploring possibilities; the implications of that literature for group decision making are unknown to me), and (iii) abandon or modify

30 1 B. Kadane

the Pareto principle (see Levi (1990) and Seidenfeld (1994)). Perhaps some exploration along each line is needed to come to a balanced view.

The history of the idea of using experts as sources of information is now clearer to me, extending both to papers by Winkler and Morris before Lindley, Tversky and Brown, and papers since by Clemen, French, Lindley, Winkler and West, among others. I thank the discussants for clarifying this matter. Pefia proposes a normal model for such amalgamation, but Winkler points to the implausibility of an assumption of conditional independence among the experts.

The view of game theory that I discuss receives kind reviews from Fishbum, French and Winkler. Additionally Parmigiani indicates that he thinks de Finetti would have agreed. However, it is my impression that my Bayesian view is less acceptable to game theorists generally than to these discussants.

No one commented on the work on sequential decisions involving several decision makers. Lindley agrees on the application of the ideas to experimental design, and points to a very interesting paper he wrote with Nozer SingpurwaUa.

The discussants have clearly accepted my invitation to join the fun; I hope that readers will also.

ADDITIONAL REFERENCES IN THE DISCUSSION

Bel 1, D. E., Raiffa, H. and Tversky, A. (1988). Decision Making: Normative, Descriptive and Prescriptive Interactions. Cambridge: University Press.

ESSRDA (1990). Expert Judgement in Risk and Reliability Analysis: Experience and Perspective. European Safety and Reliability Research and Development Associa- tion Technical Report, CEC Joint Research Centre ISPRA.

Camerer, C. and Weber M. (1992). Recent developments in modeling preferences: uncertainty and ambiguity. Journal of Risk and Uncertainty 5, 325-370.

Caro, E., Dorrdnguez, J. I. and Gir6n, E J. (1984). Compatibilidad del m6todo de DeGroot para llegar a un consenso con la f6rmula de Bayes. Trab. Estadist. 35 139-153.

Clemen, R. T. (1989). Combining forecasts: A review and annotated bibliography, Internat. J. Forecasting 5, 559-583.

Clemen, R. T. and Winkler, R. L. (1993). Aggregating point estimates: A flexible modeling approach. Management Science 39, 501-515.

Conroy, R. M. and Winkler, R. L. (1981). Informational differences between limit and market orders for a market maker. J. Financial and Quantitative Analysis la, 703-724.

Several Bayesians: A Review 3"1

Conroy, R. M. and Winkler, R. L. (1986). Market structure: The specialist as dealer and broker, J. Banking and Finance 10, 21-36.

Cooke, R. M. (1991). Experts in Uncertainty. Oxford University Press. de Finetti, B. (1978). Decisione. Enciclopedia. Torino: Einaudi. DeGroot, M. and Mortera, J. (1991). Optimal Linear Opinion Pools. Management

Science 37, 546-558. Eliashberg, J. and Winkler, R. L. (1978). The role of attitude toward risk in strictly

competitive decision-making situations. Management Science 24, 1231-1241. Eliashberg, J. and Winkler, R. L. (1981). Risk sharing and group decision making,

Management Science 27, 1221-1235. Fishburn, P. C. (1973). The Theory of Social Choice. Princeton: University Press. Fishburn, P. C. (1987). lnterprofile Conditions and Impossibility. London: ttarwood

Academic Publishers.

Fishburn, P. C. (1988). Nonlinear Preference and Utility Theory. Baltimore: John Hopkins University Press.

French, S. (1981). Consensus of Opinion. Eur. J. of Operation Research 7, 332-340. French, S. (1986). Decision Theory: an Introduction to the Mathematics of Rationality.

Chichester: Ellis Horwood.

French, S. (1992). Strategic decision analysis and group decision support. Computer Systems and Software Engineering (Dewilde, P. and Vandewalle, J. eds.). Dordrecht: Kluwer.

French, S. and Wiper, M. P. (1991). Combining expert opinions using a normal Wishart model. Tech. Rep., School of computer Studies, University of Leeds.

Gilardoni, G. L. and Clayton, M. K. (1993). On reaching a consensus using DeGroot's iterative pooling. Ann. Statist. 21, 391-401.

Harsanyi, J. C. (1955). Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. J. Political Economy 63, 309-321.

Jaffe, J. F. and Winkler, R. L. (1976). Optimal speculation against an efficient market. Journal of Finance 31, 49-61.

Kadane, J. and Larkey, P. (1982). Subjective probability and the theory of games. Management Science 28, 113-120.

Kadane, J. and Larkey, P. (1983). Reply to Professor Harsanyi. Management Science 29, 124.

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