P 302 Lect 10 Games Theory

Poli 302 Games Theory : Lecture 10 : Strategic Form games, Mixed Game Strategy, Extensive Form, Bayesian Updating. _Explanation : _Outcomes are typically the result of strategic interactions: the intersection of the actions of two or more actors. Rational choice theory uses games theory to represent strategic interactions. _ Simplification : these are not realistic assumptions, but they help simplify reality, and are therefore interesting. _ Strategic Form Prisoner’s Dilemma (PD) game . This is an example of a cooperation games and it is intended to demonstrate just how difficult cooperation can be. This representation is called the strategic form of the game. Players A/B: Cooperate or Defect . Focus on the payoffs at the strategy intersection (numbers: years in jail: lower the number the better). _We will look first at the strategic form : 2 by 2 table; Player A+B; payoffs. _Player A’s choices. _Player B’s choices. _ Nash equilibrium : stability achieved when both players no longer have an incentive to change their choice. _ Solving strategic form games: identifying the Nash Equilibrium (NE): _Determine: Dominant strategy : eliminating the dominated strategies . _Player A: Dominant strategy. _Player B: Dominant strategy. _Nash Equilibrium at intersection. _When does PD occur in international relations: _Situations where the benefits of cooperation are high and the benefits of a sucker are low. _Although two states could be better off cooperating, because they are afraid of being exploited, they forego the benefits of cooperation. _ In security , this is typified by arms racing, where mutual disarmament is a collectively and individually better outcome, the insecurity created by the inability to make promises leads both states to pursue expensive and dangerous armament. _ Stag Hunt : This is from Rousseau . You have a party of hunters, in this case, two. If they cooperate, they can catch a large stag. If they don’t cooperate, they will not be able to catch the stag. The dilemma occurs when one hunter sees a rabbit. If that hunter chases the rabbit, he eats less than if he caught the stag, and the other hunters eat nothing. _The preference here is to cooperate to capture a stag>rabbit>nothing . _Simulates situations in which sub-optimal outcomes are made more likely by the fear of abandonment . _ Chicken : Image from the 1950s: a car towards each other down a street as a test of courage. The one that drives strait and does not swerve away is seen as more courageous. Look at the game: four outcomes. _ Preferences ordering : hero>wimp>chicken>dead . _Chicken Game: simulates situations in which the cost of mutual defection is catastrophic, but exploitation pays highly. Typical of brinkmanship crises such as the Cuban Missile Crisis. 1

_There is no dominant strategy in a game with cycling strategies, so it cannot be solved using a pure game strategy. To solve a game with cycling requires a Mixed Game Strategies : Mixed Game Strategy Solution : _When a game has no dominant strategies, and there is no equilibrium from Best-Response Analysis, and there is no equilibrium from the Minimax Method, a Mixed Game Strategy solution will always exist. It consists of a probabilistic strategy in which the player plays different strategies with a certain probability. _The logic of mixed games strategy is to choose a combination of strategies that neutralizes the opponent’s ability to exploit the differences in their payoffs. If a player receives higher payoffs from one choice over another, then the opponent will offer that choice less frequently. Essentially, the mixed game strategy is neutralizing the opponent’s ability to exploit the differences in payoffs by making them indifferent between their different strategies, and this is achieved by making available the lower payoff cells more often than the higher payoff cells to the opponent. There are theorems that demonstrate that, amazingly, mixed game strategies work in both zero-sum and non-zero-sum payoff structures. _The outcome is that each player will do better playing their probabilistic strategies irrespective of what the other player does. If both players follow their optimal probabilistic mixed game strategies , then the probabilistic intersection is the mixed game strategy Nash Equilibrium . Obviously a given player will do best if they have a spy that can find out what strategy the opponent will play, but in the absence of a spy, the mixed game strategy provides the best possible payoff for the player that makes use of it, and any player that deviates from the probabilistic strategy will receive less payoffs than if they have abided by the mixed game strategy. Preference: 3>0 Player B ∂ 21 (2/3) ∂ 22 (1/3) Negot Attack Ally Surrender Negotiate Plyr A ∂ 11 (2/3) Attack ∂ 12 (1/3) Alliance Surrender  Eliminate dominated strategies (fwd and withdraw eliminated for both). 2 1 2 2 1 3 0 0 3 1 2 2 1 1 2 0 3 1 2 1 2 3 0 0 3 3 0 3 0 3 0 0 0

Player B Attack Ally Player A Attack Alliance For Player B : 1. Player A right (∂ 21 ,∂ 22 ) = ∂ 21 + 2∂ 22 (horizontal of Player’s A’s values) Player A left (∂ 21 ,∂ 22 ) = 2∂ 21 + 0∂ 22 (horizontal of Player’s A’s values) 2. ∂ 22 = 1- ∂ 21 3. ∂ 21 + 2∂ 22 = 2∂ 21 + 0∂ 22 ► ∂ 21 + 2(1-∂ 21 ) = 2∂ 21 + 0∂ 22 ► ∂ 21 + 2 - 2∂ 21 = 2∂ 21 ► 2 = 3∂ 21 ► 2/3 = ∂ 21 ; ∂ 22 = 1- ∂ 21 ; ∂ 22 = 1/3 For Player A : 1. Player B right (∂ 11 ,∂ 12 ) = 1∂ 11 + 3∂ 12 (vertical of Player’s B’s values) Player B left (∂ 11 ,∂ 12 ) = 2∂ 11 + ∂ 12 (vertical of Player’s B’s values) 2. ∂ 12 = 1- ∂ 11 3. ∂ 11 + 3∂ 12 = 2∂ 11 + ∂ 12 ► ∂ 11 + 3(1-∂ 11 ) = 2∂ 11 + 1-∂ 11 ► ∂ 11 + 3 - 3∂ 11 = ∂ 11 + 1 ► 3 - 2∂ 11 = ∂ 11 + 1 ► 2 = 3∂ 11 ► 2/3 = ∂ 11 ; ∂ 12 = 1- ∂ 11 ; ∂ 12 = 1/3  Verification For Player B: 2/3 + 2(1/3) = 2(2/3) For Player A: 2/3 + 3(1/3) = 2(2/3) + 1/3 3 2 1 1 2 1 2 3 0

Mixed Game Strategy Equations – Blank : Preference: 3>0 Player B ∂ 21 ∂ 22 CC AA BB DD CC Pl A ∂ 11 BB ∂ 12 AA DD  Eliminate dominated strategies (CC and DD eliminated for both). For Player B : 1. Player A BB (∂ 21 ,∂ 22 ) = a∂ 21 + b∂ 22 (horizontal) Player A AA (∂ 21 ,∂ 22 ) = c∂ 21 + d∂ 22 (horizontal) 2. ∂ 22 = 1- ∂ 21 3. a∂ 21 + b∂ 22 = c∂ 21 + d∂ 22 ► a∂ 21 + b(1-∂ 21 ) = c∂ 21 + d(1-∂ 21 ) For Player A : 1. Player B BB (∂ 11 ,∂ 12 ) = e∂ 11 + g∂ 12 (vertical) Player B AA (∂ 11 ,∂ 12 ) = f∂ 11 + h∂ 12 (vertical) 2. ∂ 12 = 1- ∂ 11 3. e∂ 11 + g∂ 12 = f∂ 11 + h∂ 12 ► e∂ 11 + g(1-∂ 11 ) = f∂ 11 + h(1-∂ 11 ) _Iteration : the effect of repeated play of the game: _If players received benefits, then as they peer into the future (termed the shadow of the future ), they see benefits to cooperation (accumulating across time), and are therefore more likely to cooperate. Repeated plays of the PD results in mutual cooperation. _ However : If the PD games has a definite end, then cooperation may not be the result. To determine the Nash Equilibrium, solve the very last PD game. The NE for PD is mutual defection. Then, given that the last game has resulted in joint defection, there is no shadow of the future or anticipated benefits of cooperation, the second to last game also has a NE of joint defection. This NE of defection is extended to the third to last game, fourth to last game, etc., until the very first game, which will also result in defection. _ 2 Level Games : 4 x x x x x x x x x x e a f b x x x x g c h d x x x x x x x x x x