GAMES IN EEXTENSIVE AND STRATEGIC FORM

1. GAMES IN EEXTENSIVE AND STRATEGIC FORM Topic #7

2. 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.

3. 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.

4. 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).

5. “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.

6. 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, while • the second mover gets more than his (pure strategy) minimax payoff, so • the maximin vs. minimax payoff gap is closed up in favor of the second mover.

7. 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.

8. 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.

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

10. 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: Swerve (SW) and Straight (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”).

11. 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 for tic-tac-toe has 9! = 362,880 end points. The number of strategies for each player is far larger.

12. 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).

13. 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).

14. Normal vs. Extensive Formand 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.

15. 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.

16. 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. • As previously noted, even board games as simple as 3x3 tick-tack-toe have enormous extensive forms, • But some political games can be represented and analyzed by manageably small extensive forms.

17. 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).

18. A Simple Example (cont.) • The tree diagram shows the extensive game form but not actually the game, • because it shows only outcomes, • not the payoffs the players receive from these outcomes. • This actually is an advantages because these payoffs vary by circumstances.

19. A Simple Example (cont.) • Suppose the players have these preferences: CongressPresident b+a b b q q b+a • Or suppose: CongressPresident b+a b b b+a q q

20. 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: NDemSDemRep 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.