Yet another Sheep

3 comments:

1. Justin said... 11:51

   A Game Theorist's Tale
   
   Bob Aumann and Ken Binmore are about to play a couple of rounds the 100-legged Centipede Game. Both are rational, and know the other to be rational, and know that they know this etc., i.e. common knowledge of rationality (CKR) obtains. Before play begins, a random device is used to decide who will take the role of Player I, and make the first move. Each plays by inputting their moves on a computer screen in their offices.
   
   In the first game, the random device chooses Aumann as Player I. Binmore goes back to writing his book on ethical philosophy, since he has no doubt that Aumann will conform to Theorem A in Aumann (1995) and follow his inductive strategy by choosing Down on the first move.
   
   In the second game Binmore is chosen to be Player I. Aumann goes back to his musings on risk aversion in the Talmud, since he has no doubt that Binmore will conform to Theorem A in Aumann (1995) and follow his inductive strategy by choosing Down on the first move. Then the unexpected happens. Aumann is jolted out of his Talmudic reveries by a beep from his computer telling him that it is his turn to make a move. “What’s going on here?” Aumann wonders aloud. “How could Ken have chosen Across given Theorem A in Aumann (1995)?” “Sure,” Aumann recalls, “Ken has expressed some doubts about the relevance of my Theorem A in Binmore (1996) (1997), but I know that he understands it! He knows that it shows that CKR implies that the opening move of the Centipede Game cannot be Across!” For a moment Aumann ponders his certainty of CKR, but quickly dismisses this flickering doubt. “I know that Ken is rational,” he tells himself, “and he knows that I know this, etc. And he certainly has no reason to doubt my rationality! So I am sure that CKR must still obtain. But how can CKR still obtain given that it is now my turn to make a move?”
   
   Aumann then wonders if Binmore could simply have made a mistake. But the computer version of the game they are playing requires each player to first write out the letters of their chosen action (Down or Across), and then confirm this twice by writing them out again. Aumann doesn’t think that Binmore could have written down Across three times when he really intended to write Down. “No, that also is impossible,” he tells himself, “so what is Ken doing?”
   
   Aumann then considers the only other explanation he can think of. “Ken is telling me that he is not going to follow his inductive strategy,” he decides. “Although he understands Theorem A in Aumann (1995), he is going to ignore it! But can he really do that?” Since Aumann’s first decision node has been reached, he quickly realizes that Binmore can indeed decide to ignore Theorem A. “Then why would he do it?”, he wonders. “It would only be rational if Ken thought that that I also will ignore Theorem A , but how could he predict that?” “But,” he thinks, “now that I know that Ken has not followed his inductive strategy in his first move, clearly I cannot be sure that he will follow it in his second, third or even twentieth move. So I should consider not following my inductive strategy for a number of moves also. But for how many, I wonder?”
   
   Aumann quickly decides that he has no basis for placing a probability distribution on Binmore’s moves for most of the remainder of the game. “Surely,” he thinks, “Ken will play Down at his 50th node (at vertex 99) since he can be sure that I will play Down if my 50th node is reached. So I will surely want to play Down at my 49th node, that much is clear. For how long does Ken think we can both play Across before we are too close to the end of the game?” Aumann doesn’t know the answer to that question , and realizes that he will not receive any really useful information about it as the game proceeds. He shakes his head in wonder, puts away his copy of the Talmud, and sits down at his computer screen to type in the word Across three times. A few moments later his computer beeps to tell him that it is his turn to move again.
   
   Forth-five minutes later the game ends when, at vertex 87, Binmore suddenly plays Down. Or at least it would have if Aumann hadn’t already ended the game by playing Down at vertex 78 (he is Bob Aumann after all!). Aumann pocketed a handsome payoff of $250, and Binmore received the tidy sum of $125.
   
   In the next game, it was Aumann’s turn again to start play. Binmore went back to writing his philosophy book muttering something about “old dogs” to himself. A few moments later his computer beeped….
   
   
   Reference:
   R. Aumann (1995) “Backward induction and common knowledge of rationality,” Games and Economic Behavior, 8: 6-19.  

2. Gabriel M. said... 12:17

   >> "I'll write my own take on the issue later."
   
   Now, now, now!  

3. Gabriel M. said... 13:12

   Post, post!  

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