Behavioural Economics Lecture Notes PDF

ES22018A: Behavioural Economics Chapter 5: Strategic Interaction Dr. Jörg Franke Strategic Interaction and Game Theory Overview: I Part I: Analytical Game Theory (normative) I Part II: Behavioural Game Theory (positive) 1 / 22 ...

ES22018A: Behavioural Economics Chapter 5: Strategic Interaction Dr. Jörg Franke Strategic Interaction and Game Theory Overview: I Part I: Analytical Game Theory (normative) I Part II: Behavioural Game Theory (positive) 1 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Part I Analytical Game Theory 2 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Introduction I Real-world decision problems frequently involve a strategic or interactive element I Your outcome/payoff does not only depend on your choice but also on the choice of other agents I If you are interested in own payoff maximisation, you should take into account the choice of other agents, who will do the same with you I Examples: Chess games, conflict and war, cooperation without binding commitments, Oligopolists deciding about production levels, etc. I Game Theory is the analysis of strategic interactions 3 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Analytical Game Theory: Description and Definitions I A game is a decision problem in which the final outcome depends upon the decisions of all agents, as well as the state of the world I Agents in the game are called players I A strategy is a complete plan of action that describes what a player will do under all possible circumstances I A strategy profile is a vector of strategies, one for each player I The payoff matrix represents the payoffs/utility levels associated with each possible combination of strategies (i.e. strategy profile) 4 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Exercise 5.1 I Two student go out together on the night before the exam I They oversleep, miss the exam and decide to invent the excuse of commuting together to the university and having had a flat tire on the way to university at the day of the exam I They manage to get a resit-exam I One of the questions of the resit-exam is the following: Which tire was broken? 1. Describe the exam situation as a game and identify the respective elements. 5 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Best Response and Nash Equilibrium Definition 5.1: Best Response For a specific strategy of the other players, a Best Response is a strategy that maximises own utility given the specific strategy of the other players. Analytical game theory predicts that rational players should play a specific equilibrium strategy profile: Definition 5.2: Nash Equilibrium A Nash equilibrium is a strategy profile such that each strategy in the profile is a best response to the other strategies in the profile (i.e. strategies are mutual best responses) 6 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Existence Result due to Nash (1951) Theorem 5.1: Nash Equilibrium Existence Every finite game (in which all players have a finite number of pure strategies) has a Nash equilibrium. Remarks: I There might be more than one Nash equilibrium. I The Nash equilibrium might not be in pure strategies (i.e. might involve probability distributions over more than one strategy) Exercise 5.1 continued: Find the Best Responses and the Nash equilibrium/a of the game introduced in Exercise 5.1. 7 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Some Prominent Games Exercise 5.2: Battle of the Sexes (Coordination Game) I You and your partner both like going together to concerts I While your partner prefers going to heavy metal concerts (utility level 2 for your partner and 1 for you), you prefer visiting together gangsta rap events (utility level 2 for you and 1 for your partner) I If you cannot agree and go to separate concerts, you and your partner’s utility is 0 1. Construct the payoff-matrix of this impure coordination game. 2. Find the Nash equilibria in pure strategies. 8 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Exercise 5.3: Prisoners’ Dilemma I Two criminals are arrested on suspicion of two separate crimes I Prosecutor has sufficient evidence to convict each criminal on the minor charge (2 years in jail), but not on the major charge (20 years in jail) I Prosecutor offers following deal: I If only one of the criminals declares to be culpable of the major charge (defect), then he goes free while the other serves 20 years in jail I If both declare to be culpable, they serve 10 years each in jail I If neither of the two criminals defect (they cooperate), they are only convicted of the minor charge 1. Construct the payoff-matrix of the Prisoners’ Dilemma game. 2. Find the Nash equilibria in pure strategies. 9 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Pareto Dominance and Pareto Optimality Definition 5.3: Pareto Dominance A strategy profile X Pareto dominates strategy profile Y, whenever all players weakly prefer the outcome under X over the outcome under Y. Definition 5.4: Pareto Optimality A strategy profile X is Pareto optimal if it is not Pareto dominated by any other strategy profile. Exercise 5.2, 5.3 continued: Verify whether the Nash equilibria identified in Exercise 5.2 and 5.3 are pareto-optimal. 10 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Prisoners’ Dilemma and Pareto-Optimality I In the Prisoners’ Dilemma game the Nash equilibrium is not pareto-optimal I Although rational, players cannot coordinate on efficient allocation (violation of Adam Smith’s invisible hand statement) I Question: How could rational players overcome this inefficiency: 11 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Prisoners’ Dilemma and Pareto-Optimality I In the Prisoners’ Dilemma game the Nash equilibrium is not pareto-optimal I Although rational, players cannot coordinate on efficient allocation (violation of Adam Smith’s invisible hand statement) I Question: How could rational players overcome this inefficiency: 1. Pre-play communication: Verbal agreement to ’coordinate’ not credible because ’cheap talk’-property 2. Repetition: Finite repetition involves non-credible threats to defect (backward induction), infinite repetition with sufficiently patient players (i.e. high discount factor) might involve credible threats and therefore induce the efficient outcome (but: multiple equilibria exist) 3. Modifying the payoff structure by inducing credible, strategy-dependent fines 11 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Exercise 5.4: Anti-Coordination Game I You want to have a coffee with your new boy friend. The only viable options are coffee place A or B. I Your ex-boy friend wants to talk to you because there are still some issues to settle. He will assume that you show up in coffee place A or B and will try to meet you there. I However, you prefer to avoid meeting him. I If your ex-boy friend manages to meet you, the induced utility levels are (−1, 1) I If you manage to avoid your previous boy friend, the induced utility levels are (1, −1) 1. Construct the payoff-matrix of this anti-coordination game. 2. Find the Nash equilibria in pure strategies and derive the Pareto-efficient strategy profile. 12 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Remark: The anti-coordination game from Exercise 5.4 (also called matching pennies) is a zero-sum game, where the payoffs/utilities of both players sum up to zero, regardless of the specific strategy profile. Definition 5.5: Nash Equilibrium in Mixed Strategies A Nash equilibrium in mixed strategies is a probability distribution over pure strategies such that each player is indifferent among the pure strategies in her equilibrium strategy given the other player randomizes according to her equilibrium distribution function. Exercise 5.4 continued: Find the Nash equilibrium in mixed strategies. Exercise 5.2 continued: Find the Nash equilibrium in mixed strategies. 13 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Remarks: 1. Implication of Theorem 5.1: A (finite) game might have no Nash equilibrium in pure strategies but will then have a Nash equilibrium in mixed strategies, e.g. anti-coordination game. 2. A game might have multiple Nash equilibria in pure and/or mixed strategies, e.g. battle of the sexes. 3. In a mixed strategy equilibrium a player must ’randomize’ such that the other player is indifferent among the pure strategies of her equilibrium profile ⇒ The probability distribution of a player will only depend on the payoff of the other player 14 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Exercise 5.5: Chicken (Hawk-Dove) Game I Two people drive cars at high speed in the direction of a seaside cliff I The person who brakes/stops his car first is called ’chicken’ and looses the game (payoff 1), while the other person is the winner (4) I If both drivers stop at the same time, there is a draw (2) I If both drivers do not stop, the outcome is catastrophic (-10) I Source: ’Rebel without a Cause’ starring James Dean 1. Construct the payoff-matrix of this anti-coordination game. 2. Find all the Nash equilibria and the pareto optimal strategy profiles. 15 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Chicken (Hawk-Dove) Game: Discussion I Interpretation: A resource is contested, which is rivalrous but non-excludable, moreover there is a negative externality in mutually aggressive behaviour I The Hawk-Dove Game is frequently used in political science to model conflict and military strategy I The Chicken Game is frequently used in biology to model aggressive vs. accommodating behaviour within species: I Players within one species are genetically programmed to play one of the two strategies I Players who have a higher relative payoff, replicate more due to higher fitness I The only Evolutionary Stable Strategy (ESS) is the mixed strategy Nash equilibrium ⇒ Fixed proportion of both strategies within one species 16 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement I.3 Equilibrium Refinement: Credible/Non-Credible Threats Exercise 5.6: Entry Game I There is a market with a incumbent I who obtains monopoly profit of 2 I An entrant E has to decide whether to enter in this market I If the entrant firm does not enter (Out), her profit is 2 I If the entrant firm decides to enter (In), the incumbent has two options: I Do nothing (DN), which implies that profits decline by 1, while the entrant obtains 4 I Start a price war (F), which implies that both profits fall to 0 1. Construct the payoff-matrix of this game. 2. Find all Nash equilibria in pure strategies. 3. Represent this sequential game in the extensive form. 4. Discuss the plausibility of the equilibria. 17 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Exercise 5.6: Discussion I Equilibrium (F,Out) contains a non-credible threats I Equilibrium (F,Out) entails strategies which are not Nash equilibria in all the subgames of the game Definition 5.7: Subgame-Perfect Equilibrium A subgame-perfect equilibrium is a strategy profile that constitutes a Nash equilibrium in each subgame. Existence Result due to Selten (1965) Theorem 5.3: Subgame-Perfect Equilibrium Existence Every finite extensive form game has a subgame-perfect Nash equilibrium. Remark: Subgame perfect equilibria can be identified through backward induction in the extensive form game 18 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Example 5.2: Mutually Assured Destruction (MAD) I MAD is a military doctrine from the Cold War according to which two superpowers can maintain peace by threatening each other to annihilate the human race in case of enemy attack 19 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Example 5.2: Mutually Assured Destruction (MAD) I MAD is a military doctrine from the Cold War according to which two superpowers can maintain peace by threatening each other to annihilate the human race in case of enemy attack I This involves a non-credible threat in the subgame after an enemy attack occurred I Non-credible threats can be overcome by ’binding your hands’, or implementing an automatic non-stoppable reaction (which must be public knowledge) 19 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Concept Recap I Best-Response and Nash-Equilibrium I Coordination and Anti-Coordination Games: Mixed-Strategy Nash-Equilibrium I Pareto-Optimality and the Prisoners’ Dilemma I Equilibrium Refinements: Sequential Games and Subgame-Perfection 20 / 22  I.1 Nash Equilibrium in Pure Strategies Part I: Analytical Game Theory I.2 Nash Equilibrium in Mixed Strategies References I.3 Equilibrium Refinement Experiments 5.1-5.4 Link Experiments 5.1-5.4 21 / 22  Part I: Analytical Game Theory References Literature Nash, J. (1951): “Non-Cooperative Games,” The Annals of Mathematics, 54, 286–295. Selten, R. (1965): “Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. Teil II: Eigenschaften des dynamischen Preisgleichgewichts,” Zeitschrift für die gesamte Staatswissenschaft/Journal of Institutional and Theoretical Economics, 667–689. 22 / 22