Subjectivity%20and%20Correlation - 31 1 Subjectivity and...

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31 Subjectivity and Correlation in Randomized Strategies 1 Introduction Subjectivity and correlation, though formally related, are conceptually distinct and independent issues. We start by discussing subjectivity. A mixed strategy in a game involves the selection of a pure strategy by means of a random device. It has usually been assumed that the random device is a coin flip, the spin of a roulette wheel, or something similar; in brief, an ‘‘objective’’ device, one for which everybody agrees on the numerical values of the probabilities involved. Rather oddly, in spite of the long history of the theory of subjective probability, nobody seems to have examined the consequences of basing mixed strategies on ‘‘subjective’’ random devices, i.e. devices on the probabilities of whose outcomes people may disagree (such as horse races, elections, etc.). Even a fairly superFcial such examination yields some startling results, as follows: a. Two-person zero-sum games lose their ‘‘strictly competitive’’ charac- ter. It becomes worthwhile to cooperate in such games, i.e. to enter into binding agreements. 1 The concept of the ‘‘value’’ of a zero-sum game loses some of its force, since both players can get more than the value (in the utility sense). b. In certain n -person games with n Z 3 new equilibrium points appear, whose payo¤s strictly dominate the payo¤s of all other equilibrium points. 2 Result (a) holds not just for certain selected 2-person 0-sum games, but for practically 3 all such games. Moreover, it holds if there is any area whatsoever of subjective disagreement between the players, i.e., any event in the world (possibly entirely unconnected with the game under consid- eration) for which players 1 and 2 have di¤erent subjective probabilities. The phenomenon enunciated in Result (b) shows that not only the 2- person 0-sum theory, but also the non-cooperative n -person theory is modiFed in an essential fashion by the introduction of this new kind of strategy. However, this phenomenon cannot occur 4 for 2-person games This chapter originally appeared in Journal of Mathematical Economics 1 (1974): 67–96. Reprinted with permission. 1. Example 2.1 and sect. 6. 2. Example 2.3. 3. SpeciFcally, wherever there is one payo¤ greater than the value and another one less than the value (for player 1, say). 4. Except in a very degenerate sense; see Example 2.2.
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(zero-sum or not); for such games we will show 5 that the set of equilib- rium payo¤ vectors is not changed by the introduction of subjectively mixed strategies. We now turn to correlation. Correlated strategies are familiar from cooperative game theory, but their applications in non-cooperative games are less understood. It has been known for some time that by the use of correlated strategies in a non-cooperative game, one can achieve as an equilibrium any payo¤ vector in the convex hull of the mixed strategy (Nash) equilibrium payo¤ vectors. Here we will show that by appropriate methods of correlation, even points outside of this convex hull can be achieved. 6
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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Subjectivity%20and%20Correlation - 31 1 Subjectivity and...

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