Games in the normal form- An application: “An Economic Theory of Democracy”

1. Games in the normal form- An application: “An Economic Theory of Democracy” Carl Henrik Knutsen 5/6-2008

2. * • Normal form: A way of representing games. Most appropriate for static games, but can also be used for dynamic games. These are however best represented otherwise (extensive form) • Today: Static games of complete information • Complete information: Players pay-off function are known • Can have stochastic elements in game even if complete information • Perfect info: History of game known to all players. Unproblematic in static game. • Game can be interpreted as static even if players move sequentially in real world, but no players must observe others’ moves

3. The normal form • Normal form with complete and perfect information • Contains description of • Agents • Strategies, si (contingent action plan): (s1….sn) is a strategy profile • payoffs

4. A normal form game

5. Solution concepts • Strictly and weakly dominated strategies • Iterative elimination of (strictly) dominated strategies • Set of strategies that survive this elimination are rationalizable strategies • Intuitive appeal, but weak solution concept: Many remaining strategies that might seem untractable

6. Best response and Nash-equilibrium • Best response: Strategy that gives highest pay-off (or tied), given belief about other player’s choice • In two-player games strategies are best responses if and only if they are not strictly dominated • A strategy-profile is a Nash-equilibrium if and only if each player’s prescribed strategy is a best response to other player’s strategy • “Nobody regrets choice given other player’s choice” • Stability of Nash-equilibrium: No incentives to deviate unilaterally

7. Downs (1957) • Foreword: “Downs assume that political parties and voters act rationally in the pursuit of certain clearly specified goals – it is this assumption in fact, that gives his theory its explanatory power” • A classic in political economy/political economics • “Starting point” for numerous models on party and voter behavior

8. Assumptions • Democracy with periodic re-election, freedom of speech etc • Goal: Maximize political support (votes) control government. Control of office pre se and not policy is motivation in model. Policy as mean. • Majority (party or coallition) gains government • Varying degree of uncertainty • Rational and self-interested (within limits) voters and parties

9. Analysis I • Narrow model: Two “coherent” parties or two candidates, no uncertainty, one dimension • Further assumption, voters’ ideal locations on policy-dimension (e.g. left-right) are uniformly distributed on interval. Normalize to [0,1] • Strategy sets for two candidates, S1=S2 = [0,1] • Strategy profile denoted (s1,s2) • Uniform distribution: Number (share) of voters = width of intervals. • If s2>s1 all voters to left of (s1+s2)/2 votes for 2

10. An example • 2 chooses policy 0,7, 1 chooses 0,6. All voters to the left of 0,65 votes for 21 wins. • Can this be a proper solution to the game? No! • Nash Equilibrium: Player 2 is not playing best response to 0,6. Will win majority if plays for example 0,59 • But then 1 will not play a best response..choose for example 0,58

11. Nash-equilibrium? • Assume winning government gives ui=1 and not winning gives ui=-1 • (s1,s2)= (0,5 , 0,5) is a Nash equilibrium, since both strategies are best responses to other. (Assume 50% probability of winning when tie). Nobody wants to unilaterally deviate: We have at least one NE (existence) • Any other? No! Proof by contradiction (reductio ad absurdum) If not (0,5 , 0,5), at least one has incentive to deviate. We therefore have one and only one NE (uniqueness) • EU1= EU2 = 0 in NE • Same outcome from elimination of weakly dominated strategies. 0,5 weakly dominates all other strategies. 0,5 gives equal or better result than any other strategy for a player, independent of choice of strategy for other player. • Vote maximization and winning government give identical solutions in this set-up

12. Extensions: ideological candidates • Ideological candidates: Ideal points at 0 and 1 for candidates 1 and 2 • U1 = -X2, U2 = -(1-X2) are M&M’s suggested utility functions, can be generalized to U(x) = h(-|x-z|) • NE is still (0,5, 0,5) • Logic: Whenever 1 sets policy closer to ideal point (0), 2 can obtain government and thereby a policy that is closer to the other’s ideal point (1) by setting policy closer to 0,5 than 1’s policy. E.g., 0,45 will be beaten by 0,54. 1 is worse off than if had set 0,5 as policy. • Can not be arrived at by iterated elimination of weakly dominated strategies. Why?

13. Extensions: ideological candidates with uncertainty • Ideological candidates with uncertainty (median voter on interval is random draw) • See M&M (105-107) • U1 = -X2, U2 = -(1-X2) • First, recognize that player’s will not play strictly dominated strategies: 1 will never play s1>s2 and 2 will never play s2<s1 . In fact, s1>0,5 and s2<0,5 are strictly dominated. • Then express the expected utilities of the two players: Remember that EU(p) = p1u1 + p2u2+…+pnun and n is here 2 (win and lose)

14. Continued example • EU1 = p(win)*u(win)+p(lose)*u(lose) ((s1+s2)/2)*-s12 + (1-(s1+s2)/2)*-s22 • Max expression with respect to s1. s2 is taken as given. Differentiate and set equal to 0. • After some algebra we obtain the “best response function” (*) s1 =s2/3 (the other solution is not in [0,1] • Perform identical operation for player 2: We obtain 2’s best response function (**) s2 = 2/3 + s1/3 • We now have two equations and two variables gives unique solution. Insert s2 from ** into right hand side of *  Optimal solutions are ¼ and ¾ for 1 and 2 respectively.

15. The logic • Players max expected utility • One player’s utility depends on other player’s choice of strategy We therefore obtain two general best-response functions, which indicate best response for one player given different strategy-choices by other player. • We know that in NE, both players must play best response • We are therefore in NE on a point that satisfies both BR-functions: solve as equation system..and voila (should have checked for second order conditions etc, but….)