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GAMES IN EEXTENSIVE AND STRATEGIC FORM Topic #7

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Review: A Best Reply Given a strategy choice by the other player, your best reply to it is the strategy that gives the highest payoff in that contingency. – For Player R (who is maximizing), s1 is his best reply to c1 and s2 is his best reply to c2. – For player C (who is minimizing), c2 is his best reply to s1 and c1 is his best reply to s2. – A dominant strategy is a best reply to every strategy of the other player. – In a Nash equilibrium, each player is making a best reply to the other player’s strategy.

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Sequential Non-Strictly Determined Zero-Sum Games This zero-sum payoff matrix is not “strictly determined”: – with respect to pure strategies, maximin ≤ minimax; – there is no pure-strategy Nash equilibium, and – the best strategies for both players are mixed strategies.

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Sequential Games However, if the game is played sequentially, it does seem to be “strictly determined” in a practical sense. – If Player R moves first, he can anticipate that Player C, knowing what R has done, will choose his best reply to whatever R has done. – So Player R should choose his maximin pure strategy S1 and Player C should choose his best response to S1, i.e., C2 (which is not C’s minimax strategy in the payoff matrix). – By like reasoning, if Player C moves first, C chooses his minimax strategy C1 and Player R chooses his best response to C1, i.e., S1 (which is R’s maximin strategy).

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“Look Ahead and Reason Back” – Put otherwise, the first moving player P1 should – look ahead to figure out what is the second moving player P2’ s best reply to whatever P1 chooses, and then – reason back to figure what should choose, i.e., the strategy that gives P1 the best payoff given the best reply of P2 to that strategy. – This is an example of Dixit and Nalebuf’s (p. 34) First Rule of Strategy: “Look ahead and reason back.” – This First Rule applies to sequential-move games that may be either zero-sum or non-zero-sum. – The “Ex Com” during the Cuban Missile Crisis explicitly followed this rule.

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Sequential Games (cont.) So the sequential variant of a non-strictly determined zero-sum is itself “strictly determined” in the practical sense that – both players can readily determine their best strategies, – these best strategies are pure, not mixed strategies, so – a single predictable outcome results. Moreover, in the sequential variant of non-strictly determined zero-sum games there is a clear second-mover advantage. – The first mover gets his (pure strategy) maximin payoff, but – The second mover gets more than his (pure strategy) minimax payoff, i.e., – The maximin vs. minimax payoff gap is closed up in favor of the second mover.

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Sequential Games (cont.) If a zero-sum game is strictly determined, sequential choice is no different from simultaneous choice, and – there is no (first or second) mover advantage.

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Sequential Chicken Consider the Game of Chicken player sequentially: – The best reply of the second-mover is to do the opposite of whatever the first-mover does. – So the first-mover will choose Straight and “win” the game. – Except that it is not much of a “win,” since sequential play takes the “thrill” out of the game by eliminating the risk of mutual disaster. Sequential Chicken, unlike the sequential version of a non- strictly determined zero-sum game, has a clear first-mover advantage. – The first mover get his maximum payoff while the second mover gets only his maximin payoff. – The second mover has the “last clear chance” to avoid mutual disaster.

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Sequential Chicken in Extensive Form (or a Decision/Game Tree)

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Strategies A (pure) strategy is a complete plan of action for playing a game, – where choices at any move can be made contingent on whatever information is available to the player at that move. Thus in Sequential Chicken, the first-moving player P1 has just two strategies: SW and ST. But the second-moving player P2, contrary to what the 2x2 payoff matrix suggests, actually has four strategies: – choose SW unconditionally (i.e., whatever P1 has done at the first move) [call this strategy [SW/SW]; – choose ST unconditionally (i.e., whatever P1 has done at the first move) [ST/ST]; – choose SW if P1 has chosen SW and choose ST if P1 has chosen ST [SW/ST] (“do the same as P1”); and – choose ST if P1 has chosen SW and choose SW if P1 has chosen ST [ST/SW] (“do the opposite of P1”).

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Strategies (cont.) In principle, rather than actually playing any (parlor) game, each player could write down his complete strategy and turn it in to an umpire, – who would then give each player the payoffs resulting from this strategy pair. In practice, the number of strategies to choose among, and the amount of detail required to specify any strategy, is usually beyond comprehension, – even for a game as simple as three-in-a-row tic-tac-toe*, – let alone poker, chess, etc. *The player making the first move can put his “X” in any one of 9 squares. The player making the second move can put his “0” in any one of the 8 remaining squares, etc. So the extensive form of tic-tac-toe has 9! = 362,880 end points. The number of strategies for each player is far larger.

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Sequential Chicken in Normal (or Strategic) Form Thus the expanded payoff matrix for Sequential Chicken, where Row moves first and all of Columns strategies are shown, is the 2x4 matrix above. – This is called the game in normal (or strategic) form, as opposed to extensive form. Notice that in the expanded matrix, Column has a dominant strategy, – i.e., ST/SW (choose the opposite of whatever Row has chosen).

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Sequential Chicken in Normal (or Strategic) Form Row’s best reply to Column’s strategy ST/SW is to choose ST, so the game as represented by the expanded payoff matrix can be analyzed in the same manner as a simultaneous choice game. Indeed the players can be viewed as making simultaneous choices of strategies. – Column loses no flexibility by turning in his strategy to an “umpire” in advance. – The flexibility that Column gains by moving second is “built into” his dominant strategy ST/SW (and also his dominated strategy SW/ST).

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Normal vs. Extensive Form and Information Sets Simultaneous-move games may also be displayed in extensive form, – by showing “information sets.” Ordinary (non- sequential) Chicken is shown to the right. – Two points in the tree correspond to P2’s move, but – they belong to the same information set.

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Games with Perfect Information The extensive form is most useful for specifying games with (effectively) perfect information. – Formally, this means every information set contains only one element, or put otherwise whenever a player makes any choice, he knows exactly where he is in the extensive form. – More practically, this mean all choice are made sequentially and in the open. – Games with perfect information include: tic-tac-toe, checkers, chess, and most board games Roll-call voting. – Games with imperfect information include: Bridge, poker, and most card games Voting by secret ballot.

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Backwards Induction Games with perfect information can (in principle) be solved by backwards induction, – which is a generalization of the Dixit and Nalebuf’s “Look Ahead and Reason Back” rule. – Such games are in a meaningful sense “strictly determined”; in particular, no player has reason to choose a mixed strategies. However, even board games as simple as 3x3 tick-tack-toe have enormous extensive forms, – a previously noted. But some political games can be represented and analyzed by manageably small extensive forms.

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A Simple Example Suppose Congress can pass a bill either – with an amendment (outcome b+a). – without an amendment (outcome b) or The President can then either – sign the bill (b or b+a, as the case may be) or – veto the bill, so no bill is passed at all (outcome q).

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A Simple Example (cont.) The tree diagram show the extensive game form but not actually a game, – because it shows only outcomes, – not the payoffs to the players given by these outcomes. This actually is an advantages because these payoff will vary by circumstances.

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A Simple Example (cont.) Suppose the players have these preferences: Congress President b+ab bq q b+a Now suppose: Congress President b+ab b b+a q q

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The Powell Amendment In the 1950s, Rep. Adam Clayton Powell (D-NY [Harlem]) proposed a amendment to the Federal Aid to Education Bill prohibiting federal funds from going to segregated schools There were three blocs (each less than a majority) in the House, with these preferences: NDem SDem Rep b+a b q b q b+a q b+q b What will happen? Sincere (non-strategic) voting)? – Don’t look ahead and reason back. Strategic (or “sophisticated’) voting? – Look ahead and reason back.

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