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15 Backward induction: chess, strategies, and credible threats Zermelo Theorem: Two players in game of perfect information. Finite nodes. Three outcomes: – 1 can for a win for 1 – 1 can force a tie – 2 can force a loss on 1 Example: tic tac toe – player 1 can force a tie. Example: Chess has a solution. Theorem doesn’t tell us how to play – just that there is a solution.

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Proof of Zermelo’s theorem by strong induction (if true for all k <=N, true for k+1) Let N (length of game) be 1. Player 1 is the only one who moves. The outcomes may be Win, Lose or Tie. If can win, do so. If can only tie, do so. Otherwise lose. Assume the theorem is true for all games of length less than N. Then think of first move of the game – each of whose subgames has a solution.

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A game of perfect information: each player knows whose turn it is to move and which node she is at (and how she got there). Pure strategy: a complete plan of actions (which action to take at each decision node).

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Redundant cases – finding outcomes that will never be reached

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Incredible Threat

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16. Backward Induction: Reputation and Duels We think of a 1% chance that monopolist is crazy. People generally believe monopolist is sane. BUT if he fights the first case, we have reason to believe he is crazy. But then, why wouldn't he act crazy to get people to act on this. The small possibility allows him to build a reputation. We have argued that if he acts crazy, you think he is crazy, he deters late enterers. Even the sane version of monopolist will fight (act crazy) – so there is no reason to enter. If there is an epsilon chance he is crazy, then he can deter entry by fighting (by seeming crazy).

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Lessons: Introducing small probability of player being someone else – allows us to get a very different outcome. Reputation - People with short fuses – more often get their way. Think about hostage negotiations. Idea – should never negotiate with hostages. Whole areas of economy where reputation is important. Doctors/accountants –reputation needs to be that of honest.

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Game (Duel): two volunteers. Two wet sponges. Each has one sponge. You can either throw the sponge at opponent (if hit, win the game) or you take a step forward. You can only throw once. NO dodging allowed. Different – strategic is not of form "what should I do?" It is of form "When should I do it?" P i (d) – player i's probability of hitting if i shoots at distance d.

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Fact A: If player 1 knows player 2 will not shoot the next turn, he may as well step closer as the next round he'll just be closer. Assuming no one has thrown, if player i knows at distance d that j will not throw, he should step. Fact B: If player 1 knows player 2 will shoot the next turn, he should pick his better chance of winning (his throwing or other guys missing). Prob(hit i (d)) > Prob(missing j (d-1)). Pi(d) > 1-Pj(d-1) so… Step when Pi(d) + Pj(d-1)> 1 Will be some point when this is not met. d* is first time (reading graph right to left) that the sum of the probabilities is >1 Solution – first shot should occur at d* Until get to d*, no reason to shoot – using Fact A. When get close, solve by backwards induction: at end – definitely shoot. At d=1, definitely shoot, as in the next time step the opponent will shoot and will hit. We keep going backwards. Lesson – it isn't the best or worst who should shoot – but a critical distance at which whomever turn it is at that point should shoot.

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