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

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

_For state leaders, inter-state negotiations are most often made more complicated because leaders are playing two games at once: a Prisoner’s Dilemma game against the opposing state, and a Chicken game against their own domestic political opponents. As both leaders are facing the same complexity, they are in effect three games played simultaneously, termed Two-Level Games. _A surprising finding is that the weaker a leader is with respect to their own domestic opponents, the stronger is their negotiating position in international negotiations, since they can threaten the opponent state to soften their position lest they be overthrown and no negotiated agreement results. _James Fearon, “Domestic Political Audiences and the Escalation of International Disputes,” The American Political Science Review , Vol. 88, No. 3 (Sep., 1994), pp. 577-592 : _ Audience costs are political losses of popularity politicians suffer when they break a promise they made to a public. _In a crisis, such as a brinkmanship chicken game, audience costs may push politicians to avoid compromises with an adversary. _Leaders may use audience costs to signal commitment to other states. _ Example : China often permits popular demonstrations when engaged in territorial disputes with Japan. _H.E. Goemans, War and Punishment (Princeton: Princeton UP, 2000) . (1). Democratic leaders are more vulnerable to a loss of power when they lose conflicts or suffer audience costs, but the costs of losing power are never lethal. (2). (a) Leaders of more repressive states are less likely to suffer consequences than moderately repressive states, and so are less affected by domestic costs. (b). However, very and moderately repressive states that lose both suffer greater consequences on power exit. (3). Mixed Regimes that are losing wars increase their war demands as part of a gamble for resurrection. Bayes’ Theorem versus Human Intuition : 1% of people will get cancer. 99% of people will not get cancer. A test has a 90% chance of detecting cancer when it is there. 10% of cancers are missed by the test. 5% of the tests detect cancer when it is not there (95% correctly return a negative result when there is no cancer). What is your chance of having cancer? Cancer Not Cancer Test Positive 90% 5% Test Negative 10% 95% 5

_ Nature may be attributed a decision node to model instances in which one player has uncertainty about the nature of the opponent. In the model presented here, nature represents the uncertainty node for the American decision-makers of whether they are dealing with softline or hardline Soviet decision-makers during the Cuban Missile Crisis, a distinction that has serious repercussions. The undifferentiable decision nodes are enclosed in a single information set . _To identify the NE, multiply each of the payoffs by their probability (depending on their particular path), and solve by reverse induction. Extensive Form: Hostage Taking : _ Game : 2 Chance plays first determining whether terrorists are soft-line or hard-line. Militants then decided whether to take hostages or not, then the government chooses whether to negotiate with the terrorists or not, and then the terrorists choose whether to release the hostages or not. _Here we have a probability of 0.25 of the terrorists being hard-line and 0.75 of the terrorists being soft-line. _Solving the game consists first of multiplying the payoffs by their probabilities, and then solving through reverse induction. The terrorists must first decided whether to release the hostages or not. The Government must then decided whether to pay or not, without knowing whether the terrorists are soft-line or hard-line. Since the payoffs have been multiplied by the probabilities of bring soft or hard-line, the Government payoffs are simply the average of the decision to Pay (-1/2-1.5 = -2/2 = -1), versus the decision to Not Pay (-3/4-3/4 = -1.5/2 = -3/4). The government therefore chooses not to pay. Consequently, the terrorists in both instances choose not to take hostages. Extensive Form Probabilities : _Definition of Terms : | = Given; ⌐ = Not. _ Example : Given the following probabilities: 90% of crises are brinkmanship crises ( B ), and 10% are justification of war crises ( J ). Brinkmanship crises escalate to war 30% of the time (and remain as crises 70% of the time), whereas justification of war crises escalate to war 75% of the time (and remain as crises 25% of the time). _ Question 1 : What is the probability of war ( W )? _ Procedure : Map out the causality chain on the extensive form game. _ Formal Equation : P(W) = P(W|B)*P(B) + P(W|J)*P(J). _ Defined Equation : Probability of War = Probability of War Given a Brinkmanship Crisis times Overall Probability of a Brinkmanship Crisis plus Probability of War Given a Justification of War Crisis times the Overall Probability of a Justification of War Crisis. _ Calculation : P(W) = 0.3*0.9 + 0.75*0.1 = 0.27 + 0.075 = 0.345, or 34.5% chance of war. _ Question 2 : What is the probability of ongoing crisis ( C )? _ Procedure : Map out the causality chain on the extensive form game. _ Formal Equation : P(W) = P(C|B)*P(B) + P(C|J)*P(J). _ Defined Equation : Probability of Crisis = Probability of Crisis Given a Brinkmanship Crisis times the Overall Probability of a Brinkmanship Crisis plus the Probability of a Crisis Given a Justification of War Crisis times the Overall Probability of a Justification of War Crisis. 2 MacArtan Humphreys, Political Games (New York: W.W. Norton, 2017), xx 7

_ Calculation : P(W) = 0.7*0.9 + 0.25*0.1 = 0.63 + 0.025 = 0.655, or 65.5% chance of ongoing crisis. _The probability of war (34.5%) + probability of ongoing crisis (65.5%) = 100%. _ Applications of Bayesian Conditional Probabilities : _ Question 3 : Given that a war ( W ) has occurred, what is the probability that the war was caused by a brinkmanship crisis ( B ), rather than a justification of war crisis ( J )? _ Formal Equation : P(B|W) = P(W|B)*P(B)/P(W). _ Defined Equation : Probability of a Brinkmanship Crisis Given War = Probability of a War Given a Brinkmanship Crisis times Probability of a Brinkmanship Crisis divided by the Probability of War, where the “Probability of a War Given a Brinkmanship Crisis” is the subset probability for Brinkmanship that a war will result. _ Calculation : P(B|W) = (0.3 * 0.9) / .345 = 0.78, or 78% chance of the war having been caused by a brinkmanship crisis. _ Question 4 : Given that a war ( W ) has occurred, what is the probability that the war was caused by a justification of war crisis ( J ), rather than a brinkmanship crisis ( B )? _ Formal Equation : P(J|W) = P(W|J)*P(J)/P(W). _ Defined Equation : Probability of a Justification of War Crisis Given War = Probability of a War Given a Justification of War Crisis times Probability of a Justification of War Crisis divided by the Probability of a Justification of War Crisis, where the “Probability of a War Given a Justification of War Crisis” is the subset probability for Justification of War Crisis that a war will result. _ Calculation : P(J|W) = (0.75 * 0.1) / .345 = 0.21, or 21% chance of the war having been caused by a brinkmanship crisis. _The probability of war caused by Brinkmanship Crises (78%) + probability of war caused by a Justification of war Crisis (21%) = 100%. Bayesian Theorem (aggregated): 3 P(B|W) = P(W|B) * P(B) / P(C|B)*P(B) + P(C|⌐B)*P(⌐B). _ Divide the Dollar Game : _The divide the dollar game is a game demonstrating the costs of war in bargaining. Two adversaries have the choice between bargaining an outcome, meaning, negotiating and agreeing to split a dollar between themselves, or going to war over the division of a dollar. These two options are calculated as follows: _ Negotiated Solution : 100 cents is split as desired, or split evenly, 50 cents each. _ War Solution : War has two aspects: it is costly and it is unpredictable. _Step 1: D100 to determine how much side A wins. B wins 100-(D100 #). _Step 2: Each side subtracts the costs of war from their gains. Cost of war: 10 cents for each side. 3 Bayesian Calculator: http://stattrek.com/online-calculator/bayes-rule-calculator.aspx 8

Arms Race Game : 1. This game is made up of two actors, actor A and actor B. 2. Each turn in this infinite repeated horizon game, players A and B get to simultaneously apportion their state resources between negotiations and arms procurements . The total value of this state resource is 1, and may be apportioned as a whole (1), as a half (0.5), or as none (0), such that the sum is equal to one (1). 3. Player strategy form choices: choose between three levels of: negotiations {0, ½, 1} PLAYER B Negotiations (1) Negotiations (1/2) Negotiations (0) PLAYER A Negotiations (1) 1 ½ ½ 1 0 0 Negotiations (1/2) 0 1 ½ ½ 1 0 Negotiations (0) 0 0 0 ½ ½ 0 4. Player Payoffs: For ease of analysis, the payoffs will be generalized to 1 for the attainment of infinite peace, ½ for a war in which one state has military superiority, and 0 for a war in which one state is equal to or inferior to its opponent’s military force. 5. Preference Orderings: (a). Being positionally defensive, players A and B prefer peace to war (Peace > War), but if war, war with the greatest amount of arms (War(M A ) > War(M B )). (b). War with an equality in arms is the worst outcome since it is equated with an unending war of attrition, whose consequences are worse than individual victory. _ The Sequential Calculations : _War will occur whenever state A (or B) perceives that its chances of prevailing defensively becomes equal to or less than 5 per cent. In effect, a state will never take advantage of its military superiority, but will act on the basis of the implications of its future vulnerability. _The chance of war is determined individually for each player and is a function of the following sets of equations. First, the probability of Victory in War to a given player is one-tenth the division of the given player’s arms divided by the adversary’s arms, minus one one-hundredth of the current tension index: P(Victory A ) = (1/10)(Arms A /Arms B ) - (1/100)(Tension A ) P(Victory B ) = (1/10)(Arms B /Arms A ) - (1/100)(Tension B ) where 10

Arms A =  n i=3* arms procurements A Arms B =  n i=3* arms procurements B *=depreciation value as follows: arms procurement at time t multiplied by 1.0 arms procurement at time t-1 multiplied by 0.5 arms procurement at time t-2 multiplied by 0.3 and Tension A : Shock A t-1 + Militarization A t-1 + Militarization B t-1 Tension B : Shock B t-1 + Militarization B t-1 + Militarization A t-1 where Shock A = Tension A t-1 - negotiations A - negotiations B + Random Shock B = Tension B t-1 - negotiations B - negotiations A + Random and Militarization A = arms procurements A Militarization B = arms procurements B Aggregated, the equation would look like this: War A if 0.5 > (1/10)*[{( Arms A t-3 *0.3)+( Arms A t-2 *0.5)+( Arms A t-1 )}/ {( Arms B t-3 *0.3)+( Arms B t-2 *0.5)+( Arms B t-1 )}] -{[1/10]*[Tense A t-1 +Random t-1 + Negotiations A t-1 + Negotiations B t-1 + Arms A + Arms B ]}; Peace otherwise. Play of the Game : The game continues for an indefinite period of turns until war occurs. War occurs when either player’s probability of victory is equal to or less than 5 percent. During each turn, player A and B each select how they divide their state resources between arms procurements and negotiations , assigning a single unit between the two activities. The default selection, and the distribution of resources on the first turn is ½ to arms procurements and ½ to negotiations . When militarize is held constant, it assumes the value of 0.9 (which is the depreciation value of arms procurements valued at 0.5 for three consecutive turns). 11

Side B 6 6 1.3 0.5 0 0.03 0.5 WAR Round 10 Side A 7 5 1.05 0.5 0 0.04 0.5 WAR Side B 7 5 1.05 0.5 0 0.04 0.5 WAR Round 11 Side A 6 6 0.9 0.5 0 0.03 0.5 WAR Side B 6 6 0.9 0.5 0 0.03 0.5 WAR Round 12 Side A 7 5 0.9 0.5 0 0.04 0.5 WAR Side B 7 5 0.9 0.5 0 0.04 0.5 WAR Round 13 Side A 6 6 0.9 0.5 0 0.03 0.5 WAR Side B 6 6 0.9 0.5 0 0.03 0.5 WAR Round 14 Side A 7 5 0.9 0.5 0 0.04 0.5 WAR Side B 7 5 0.9 0.5 0 0.04 0.5 WAR Round 15 Side A 6 6 0.9 0.5 0 0.03 0.5 WAR Side B 6 6 0.9 0.5 0 0.03 0.5 WAR Round 16 Side A 7 5 0.9 0.5 0 0.04 0.5 WAR Side B 7 5 0.9 0.5 0 0.04 0.5 WAR Round 17 Side A 6 6 0.9 0.5 0 0.03 0.5 WAR Side B 6 6 0.9 0.5 0 0.03 0.5 WAR Round 18 Side A 7 5 0.9 0.5 0 0.04 0.5 WAR Side B 7 5 0.9 0.5 0 0.04 0.5 WAR Round 19 Side A 6 6 0.9 0.5 0 0.03 0.5 WAR Side B 6 6 0.9 0.5 0 0.03 0.5 WAR Round 20 Side A 7 5 0.9 0.5 0 0.04 0.5 WAR Side B 7 5 0.9 0.5 0 0.04 0.5 WAR Round 21 Side A 6 6 0.9 0.5 0 0.03 0.5 WAR Side B 6 6 0.9 0.5 0 0.03 0.5 WAR Round 22 Side A 7 5 0.9 0.5 0 0.04 0.5 WAR Side B 7 5 0.9 0.5 0 0.04 0.5 WAR Round 23 Side A 6 6 0.9 0.5 0 0.03 0.5 WAR Side B 6 6 0.9 0.5 0 0.03 0.5 WAR Round 24 Side A 7 5 0.9 0.5 0 0.04 0.5 WAR Side B 7 5 0.9 0.5 0 0.04 0.5 WAR 13

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