Arbitrage, incomplete models, and interactive rationality

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Acknowledgements: I am grateful to Bob Clemen, Bart Lipman, Mark Machina, Peter Wakker, and Andy Yates for helpful comments, but they are not responsible for the consequences. This research is supported by the National Science Foundation under grant 98-09225 and by the Fuqua School of Business.


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1 The terminology used here is slightly non-standard. Normally the term “event” refers to a set of states of nature, and the term “outcome” is sometimes synonymous with consequence. Here the term “event” refers generically to a set of states of nature and/or choices by human agents, and an outcome is an atomic event—e.g., a single cell in a contingency table. For an agent, every outcome is mapped to a known consequence, but different outcomes might, in principle, lead to the “same” consequence for that agent. The generic use of the term “event” admits the possibility that the objects of one agent’s uncertainty may be acts of other agents as well as acts of nature. Meanwhile, the term “alternative” is used here to refer to what is sometimes called a “concrete act,” i.e., a feasible object of choice for a human agent.

2 Nature also “behaves” by choosing among the states, but nature’s moves are modeled differently than those of the human agents. In the alternative theory introduced in section 3, states of nature and alternatives of agents are treated more symmetrically.

3 The independence axiom has been the object of much controversy, and relaxations of it have led to many varieties of “non-expected utility” theory—e.g. Machina (1982), Chew (1983), Quiggin (1982), Yaari (1987), Schmeidler (1989).

4 These labels for Savage’s axioms P3 and P4 are due to Shafer (1986), who points out that they are actually more important than the more famous independence axiom because of the strain they place on the notion of a “small world.”

5 The axiom of rational choice is not always listed as a formal assumption. Sometimes it is merely remarked that “choices follow preferences” or “preferences are revealed by choices,” or else it is assumed that preference and choice are synonymous: the agent’s “choice function” is simply given in lieu of a preference ordering. The rational choice axiom does explicitly appear in Luce and Raiffa (1957) as the “law of behavior”: assumption ix in Chapter 3, and it is informally used by Savage. Here we maintain a careful distinction between “preference” and “choice” to call attention to the question of which choices are actually observed and which are not: a “preference” is a choice that may be (and usually is) merely hypothetical because the alternatives in question are never actually faced together (and are often nonexistent or even logically impossible).

6 We will show later that the problem of one body and the problem of two or more bodies can be solved with the same mathematics after all: there is really no essential difference between individual and strategic rationality.

7 Note that Left is a dominant—i.e., always better—strategy for type A of player 2 and Right is a dominant strategy for type B. If player 2’s type is revealed to player 1 before both players move, then player 1 will play Top when 2 is type A (correctly anticipating that 2 will play Left) and 1 will play Bottom if 2 is type B (correctly anticipating a move of Right). In the incomplete-information version, player 2 benefits from player 1’s uncertainty and is able to obtain a higher payoff when her type is B.

8 Models of rational asset pricing have been enormously successful in finance, but the most robust and durable asset pricing models rely on arbitrage arguments and statistical regularities rather than detailed models of behavior at the agent level.

9 The role-playing agent is sometimes referred to as homo sociologicus, in contrast to homo economicus. The latter seeks to satisfy her own preferences, the former merely attends to her “station and its duties.” (Heap et al. 1992)

10 It is actually no longer necessary to distinguish a priori between events controlled by one player and events controlled by another, nor, indeed, between alternatives and states of nature. The question of who controls what need not be answered in order for the behavior of the players to be judged rational. What matters is the way in which agents are willing to bet on the outcomes of events, which may reveal what they believe about who is in control. The usual terminology nevertheless will be retained for convenience.

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