Appendix: Rationalizability (Advanced) 85
the game that is derived by deleting strictly dominated strategies from the original game. Continuing in this manner, we derived the strategies that sur- vive the IDSDS.
As just described, the IDSDS eliminates what players would not do if ra- tionality were common knowledge. But what is it that they would do? If, after using the IDSDS, there is only one strategy remaining for each player, then this procedure gives us a clear and definitive description as to how players will behave. For once all that is “impossible” has been eliminated, then “whatever remains, however improbable, must be the truth.” But suppose multiple strategies survive the IDSDS? Although we eliminated what is inconsistent with rationality being common knowledge, is all that remains consistent with rationality being common knowledge?
Remember that rationality means acting optimally, given one’s beliefs about what other players will do. Thus, a strategy is consistent with rational- ity only if at least some beliefs about the other players’ strategies make that strategy the best one. Let’s try working directly with that definition and see what happens.
Consider a two-player game. If player 1 is rational, then she chooses a strat- egy that maximizes her payoff, given her beliefs as to the strategy of player 2. But what are reasonable beliefs for player 1 to hold about player 2’s strategy? If player 1 believes that player 2 is rational, then she will expect player 2 to use a strategy that maximizes his payoff, given his beliefs about her strategy. Should we allow player 1 to expect that player 2 would hold just any old be- liefs as to what player 1 will do? Not if rationality is common knowledge. If player 1 believes that player 2 believes that player 1 is rational, then player 1 believes that player 2’s beliefs about player 1’s strategy ought to be consistent with player 2’s believing that player 1 is rational, which means that player 2 believes that player 1 plays a strategy that is optimal for some beliefs about 2’s strategy. And of course, it doesn’t end there, but since my head is starting to hurt, let’s jump to a more general statement.
✚ DEFINITION 3.4 A strategy is rationalizable if it is consistent with ra- tionality being common knowledge, which means that the strategy is optimal for a player, given beliefs that are themselves consistent with rationality being common knowledge
.
That’s not a user-friendly definition, so my plan of explaining rationaliz- ability with more generality may have backfired. So let’s move in the other direction and work with a particular example. Consider the game shown in FIGURE A3.1.
Think about determining whether strategy a is rationalizable for player 1. Are there beliefs about what player 2 will do that would make a optimal for player 1? Yes, since a is best if and only if player 1 believes that player 2 will use x. But does player 2 have beliefs about what player 1 will do that makes it optimal for player 2 to use x? If not, then it doesn’t make much sense for player 1 to believe that player 2 will use x (since player 1 believes that player 2 is rational), and without such a belief, there’s not much of an ar- gument for player 1 to use a. In fact, x is optimal for player 2 if and only if player 2 believes that player 1 will use b.
1,3
3,3
5,0
2,0
3,1
1,2
1,1
2,0 Player 1
Player 2
a
b
c
d
1,2
4,2
3,1
0,1
yx z
FIGURE A3.1 Solving a Game for the Rationalizable Strategies
Let’s summarize thus far: It makes sense for player 1 to use a if she believes that player 2 will use x. It makes sense for player 1 to believe that player 2 will use x if player 2 believes that player 1 will use b. But then, this just begs another question: Is it reasonable for player 2 to believe that player 1 will use b? If b is a poor strategy for player 1, then the belief supporting player 2’s using x is under- mined and, with it, the argument for player 1 to use a. In fact, b is strictly dom- inated by c for player 1. If player 1 believes that player 2 believes that player 1 is rational, then player 1 should not believe that player 2 believes that player 1 will use b, and thus we cannot rationalize player 2’s using x and thus cannot ra- tionalize player 1’s using a. Strategy a is not rationalizable, because it is not op- timal for player 1 on the basis of beliefs that are themselves consistent with ra- tionality being common knowledge.
Now consider strategy c, and let us argue that it is rationalizable. Strategy c is optimal for player 1 if she believes that player 2 will use z. But is z opti- mal for player 2, given some beliefs about what player 1 will do? We need that to be the case in order for player 1 to believe that player 2 will use z, because, recall that player 1 believes that player 2 is rational and rational players only use a strategy that is optimal, given their beliefs. If player 2 believes that player 1 will use d, then playing z is indeed best for player 2. But is it reason- able for player 2 to believe that player 1 will use d? Yes it is, because d is opti- mal if player 1 believes that player 2 will play y. Hence, it is reasonable for player 1 to believe that player 2 believes that player 1 will play d when player 1 believes that player 2 believes that player 1 believes that player 2 will play y. But is that belief consistent with rationality being common knowledge? (You might think that this could never end, but just hang in there for one more round of mental gymnastics.) If player 2 believes that player 1 will play c, then playing y is optimal for player 2. Hence, it is reasonable for player 1 to believe that player 2 believes that player 1 believes that player 2 will play y when player 1 believes that player 2 believes that player 1 believes that player 2 be- lieves that player 1 will play c. Now, what about that belief? Well take note that we’re back to where we started from, with player 1 playing c. We can then re- peat the argument ad infinitum:
1. Player 1’s playing c is optimal when player 1 believes that player 2 will play z.
2. Player 2’s playing z is optimal when player 2 believes that player 1 will play d.
3. Player 1’s playing d is optimal when player 1 believes that player 2 will play y.
4. Player 2’s playing y is optimal when player 2 believes that player 1 will play c.
5. Player 1’s playing c is optimal when player 1 believes that player 2 will play z.
6. Repeat steps 2–5.
After intense use of our “little gray cells” (as the detective Hercule Poirot would say), we conclude that strategy c is rationalizable for player 1 because it is optimal for player 1 given beliefs as to what 2 will do and those beliefs are con- sistent with rationality being common knowledge. Furthermore, all strategies in
86 CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
References 87
that cycle are rationalizable using those beliefs. For example, z is optimal for 2 if 2 believes 1 will use d, and 1 using d is optimal if 1 believes 2 will use y, and y is optimal for 2 if 2 believes 1 will use c, and 1 using c is optimal if 1 believes 2 will use z, at which point we’re back where we started from. Hence, strategies c and d are rationalizable for player 1 and y and z for player 2. In fact, one can show that these are the only rationalizable strategies.
If you were to apply the IDSDS to this game, you’d find that those strate- gies which survive the IDSDS are exactly the same as the rationalizable strategies just derived. Interesting? Coincidence? Not quite. First note that a rationalizable strategy also survives the IDSDS because being rational implies not using a strictly dominated strategy. But can a strategy survive the IDSDS and not be rationalizable? Yes, it is possible, although the technical nature of that difference is not one that will concern us in this book. Furthermore, in a wide class of circumstances, the two concepts deliver the same answer. As you can imagine, the IDSDS is vastly easier to understand and use, which are good enough reasons for me to make it the focus of our attention. Nevertheless, it is important to keep in mind that it is the concept of rationalizability that directly encompasses what it means for a strategy to be consistent with ra- tionality being common knowledge.
REFERENCES 1. This application is based on Eugene Smolensky, Selwyn Becker, and
Harvey Molotch, “The Prisoner’s Dilemma and Ghetto Expansion,” Land Economics, 44 (1968), 419–30.
2. This game is from Steven Brams, Superior Beings (New York: Springer- Verlag, 1983). Brams has established himself as one of the most creative users of game theory.
3. B. A. Baldwin and G. B. Meese, “Social Behaviour in Pigs Studied by Means of Operant Conditioning,” Animal Behaviour, 27 (1979), 947–57.
4. See, for example, Edward J. Bird and Gert G. Wagner, “Sport as a Common Property Resource: A Solution to the Dilemmas of Doping,” Journal of Conflict Resolution, 41 (1997), 749–66.
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