ABSTRACT

In this chapter we discuss the sensitivity of the minimax theorem to the cardinality of the set of pure strategies. In this light, we examine an infinite game due to Wald and its solutions in the space of finitely additive (f.a.) strategies.

Finitely additive joint distributions depend, in general, upon the order in which expectations are composed out of the players' separate strategies. This is connected to the phenomenon of “non-conglomerability” (so-called by deFinetti), which we illustrate and explain. It is shown that the player with the “inside integral” in a joint f.a. distribution has the advantage.

In reaction to this asymmetry, we propose a family of (weighted) symmetrized joint distributions and show that this approach permits “fair” solutions to fully symmetric games, e.g., Wald's game. We develop a minimax theorem for this family of symmetrized joint distributions using a condition formulated in terms of a pseudo-metric on the space of f.a. strategies. Moreover, the resulting game can be solved in the metric completion of this space. The metrical approach to a minimax theorem is contrasted with the more familiar appeal to compactifications, and we explain why the latter appears not to work for our purposes of making symmetric games “fair.” We conclude with a brief discussion of three open questions relating to our proposal for f.a. game theory.

INTRODUCTION

In this essay we derive results for finitely additive (mixed) strategies in two-person, zero-sum games with bounded payoffs.

ABSTRACT

For Savage (1954) as for deFinetti (1974), the existence of subjective (personal) probability is a consequence of the normative theory of preference. (DeFinetti achieves the reduction of belief to desire with his generalized Dutch-Book argument for previsions.) Both Savage and deFinetti rebel against legislating countable additivity for subjective probability. They require merely that probability be finitely additive. Simultaneously, they insist that their theories of preference are weak, accommodating all but self-defeating desires. In this chapter we dispute these claims by showing that the following three cannot simultaneously hold:

1. i. Coherent belief is reducible to rational preference, i.e. the generalized Dutch- Book argument fixes standards of coherence.

2. ii. Finitely additive probability is coherent.

3. iii. Admissible preference structures may be free of consequences, i.e. they may lack prizes whose values are robust against all contingencies.

I. INTRODUCTION

One of the most important results of the subjectivist theories of Savage and deFinetti is the thesis that, normatively, preference circumscribes belief. Specifically, these authors argue that the theory of subjective probability is reducible to the theory of reasonable preference, i.e. coherent belief is a consequence of rational desire. In Savage's (1954) axiomatic treatment of preference, the existence of a quantitative subjective probability is assured once the postulates governing preference are granted. In deFinetti's (1974) discussion of prevision, avoidance of a (uniform) loss for certain is thought to guarantee agreement with the requirements of subjective probability (sometimes called the avoidance of “Dutch Book”).

In the seventeen years between the first (1954) and second (1971) editions of his book The Foundations of Statistics, Savage reported a change in the“climate of opinion” about foundations. That change, he said,“would obliterate rather than restore” his earlier thinking about the relationship between the two major schools of statistics that were the subject of his inquiry. What in the early 1950s started out for Savage as a task of building Bayesian expected utility foundations for common, so-called Frequentist statistics – which for Savage included the Fisher-Neyman-Pearson-Wald program that was dominant in the British-American school from the 1930s – revealed itself, instead, to be an impossibility. Contrary principles separated Bayesian decision theory from what practicing statisticians of the day were taught to do. Significance tests, tests of hypotheses, and confidence intervals give quantitative indices, such as confidence levels, that only accidentally and approximately cohere with the Bayesian theory that Savage hoped might elucidate and justify them.

Since the second edition of The Foundations of Statistics and Savage's premature death (both in 1971), we have come to understand much better the extent of the conflict between Bayesian and Frequentist statistical principles of evidence. Some limitations in Frequentist methods, which existed only in the lore of practicing statisticians, gained theoretical footing through the Bayesian point of view. Consider the very general problem of how, within the Frequentist program, to deal with conditional inference, e.g., whether or not to condition on an ancillary statistic.

Randomization has thus been a puzzle for Bayesian theory for many years. In this essay, we give our current views on this subject.

There are two principal arguments for randomization that we are familiar with. The first is to support a randomization-analysis of the data. This notion goes back to Fisher, and is exposited in a series of papers by Kempthorne (1955, 1966, 1977). It asks whether what is observed is surprising given all the other designs that might have been randomly selected and data that might have been observed, but were not. By its appeal to what did not occur, such an analysis violates the likelihood principle; hence, it is not compatible with Bayesian ideas.

ABSTRACT

This investigation combines two questions for expected utility theory:

1. When do the shared preferences among expected utility maximizers conform to the dictates of expected utility?

2. What is the impact on expected utility theory of allowing preferences for prizes to be state-dependent?

Our principal conclusion (Theorem 4) establishes very restrictive necessary and sufficient conditions for the existence of a Pareto, Bayesian compromise of preferences between two Bayesian agents, even when utilities are permitted to be state-dependent and identifiable. This finding extends our earlier result (Theorem 2, 1989a), which applies provided that all utilities are stateindependent. A subsidiary theme is a decision theoretic analysis of common rules for “pooling” expert probabilities.

Against the backdrop of “horse lottery” theory (Anscombe and Aumann 1963) and subject to a weak Pareto rule, we show, generally, that there is no Bayesian compromise between two Bayesian agents even when state-dependent utilities are entertained in an identifiable way. The word “identifiable” is important because if state-dependence is permitted merely by dropping the Anscombe-Aumann axiom (Axiom 4 here) for “state-independence,” though a continuum of possible Bayesian compromises emerges, also it leads to an extreme underdetermination of an agent's personal probability and utility given the agent's preferences. Instead, when state-dependence is monitored through (our version of) the approach of Kami, Schmeidler, and Vind (1983), the general impossibility of a Bayesian, Pareto compromise in preferences reappears.

ABSTRACT

When can a Bayesian investigator select an hypothesis H and design an experiment (or a sequence of experiments) to make certain that, given the experimental outcome(s), the posterior probability of H will be lower than its prior probability? We report an elementary result which establishes sufficient conditions under which this reasoning to a foregone conclusion cannot occur. Through an example, we discuss how this result extends to the perspective of an onlooker who agrees with the investigator about the statistical model for the data but who holds a different prior probability for the statistical parameters of that model. We consider, specifically, one-sided and two-sided statistical hypotheses involving i.i.d. Normal data with conjugate priors. In a concluding section, using an “improper” prior, we illustrate how the preceding results depend upon the assumption that probability is countably additive.

EXPECTED CONDITIONAL PROBABILITIES AND REASONING TO FOREGONE CONCLUSIONS

Suppose that an investigator has his or her designs on rejecting, or at least making doubtful, a particular statistical hypothesis H. To what extent does basic inductive methodology insure that, without violating the total evidence requirement, this intent cannot be sure to succeed? We distinguish two forms of the question:

1. Can the investigator plan an experiment so that he or she is certain it will halt with evidence that disconfirms W.

2. Can the investigator plan an experiment so that others are certain that the investigator will halt with evidence that disconfirms H?

ABSTRACT

We review the axiomatic foundations of subjective utility theory with a view toward understanding the implications of each axiom. We consider three different approaches, namely, the construction of utilities in the presence of canonical probabilities, the construction of probabilities in the presence of utilities, and the simultaneous construction of both probabilities and utilities. We focus attention on the axioms of independence and weak ordering. The independence axiom is seen to be necessary to prevent a form of Dutch Book in sequential problems.

Our main focus is to examine the implications of not requiring the weak order axiom. We assume that gambles are partially ordered. We consider both the construction of probabilities when utilities are given and the construction of utilities in the presence of canonical probabilities. In the first case we find that a partially ordered set of gambles leads to a set of probabilities with respect to which the expected utility of a preferred gamble is higher than that of a dispreferred gamble. We illustrate some comparisons with theories of upper and lower probabilities. In the second case, we find that a partially ordered set of gambles leads to a set of lexicographic utilities, each of which ranks preferred gambles higher than dispreferred gambles.

I. INTRODUCTION: SUBJECTIVE EXPECTED UTILITY [SEU] THEORY

The theory of (subjective) expected utility is a normative account of rational decision making under uncertainty.

ABSTRACT

We investigate conditions under which conditional probability distributions approach each other and approach certainty as available data increase. Our purpose is to enhance Savage's (1954) results, in defense of “personalism”, about the degree to which consensus and certainty follow from shared evidence. For problems of consensus, we apply a theorem of Blackwell and Dubins (1962), regarding pairs of distributions, to compact sets of distributions and to cases of static coherence without dynamic coherence. We indicate how the topology under which the set of distributions is compact plays an important part in determining the extent to which consensus can be achieved. In our discussion of the approach to certainty, we give an elementary proof of the Lebesgue density theorem using a result of Halmos (1950).

INTRODUCTION

In his classic discussion of Bayesian inference, L. J. Savage (1954, Sections 3.6 and 4.6) illustrates how (finitely many) different investigators come to agree on the truth of one hypothesis, given an increasing sequence of shared observations. More precisely, Savage's result is this. Assume the following two conditions.

1. The agents' initial opinions over a (finite) set of rival hypotheses are not too discrepant – there is agreement on which hypotheses have probability 0.

2. There is agreement also on the (distinct) likelihoods for these hypotheses over an infinite sequence of observations which are identically, independently distributed given an hypothesis.