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

* 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

The normal form Normal form with complete and perfect information Contains description of – Agents – Strategies, s i (contingent action plan): (s 1 ….s n ) is a strategy profile – payoffs

A normal form game Player 1\Player 2LeftRight Up(2,0)(3,1)(3,1) Down(5,2)(5,2)(2,4)

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

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

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

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

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, S 1 =S 2 = [0,1] Strategy profile denoted (s 1,s 2 ) Uniform distribution: Number (share) of voters = width of intervals. If s 2 >s 1  all voters to left of (s 1 +s 2 )/2 votes for 2

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

Nash-equilibrium? Assume winning government gives u i =1 and not winning gives u i =-1 (s 1,s 2 )= (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) EU 1 = EU 2 = 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

Extensions: ideological candidates Ideological candidates: Ideal points at 0 and 1 for candidates 1 and 2 U 1 = -X 2, U 2 = -(1-X 2 ) 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?

Extensions: ideological candidates with uncertainty Ideological candidates with uncertainty (median voter on interval is random draw) See M&M ( ) U 1 = -X 2, U 2 = -(1-X 2 ) First, recognize that player’s will not play strictly dominated strategies: 1 will never play s 1 >s 2 and 2 will never play s 2 0,5 and s 2 <0,5 are strictly dominated. Then express the expected utilities of the two players: Remember that EU(p) = p 1 u 1 + p 2 u 2 +…+p n u n and n is here 2 (win and lose)

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

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….)