Lecture 3. Offense vs. Defense & Dynamic Games EC DD & EE / Manove Offense vs Defense p EC DD & EE / Manove Clicker Question p
Using Game Theory to Analyze Offense versus Defense In many competitive situations the offense of one competitor battles the defense of the other. If the defense matches the offense, then the defense wins. If not, the offense wins. EC DD & EE / Manove Offense vs Defense p 3 Example: Military Strategies Example: Business Strategies EC DD & EE / Manove Offense vs Defense>Examples p 4
EC DD & EE / Manove Matching Pennies Matching pennies is a game-theory model of offense-versus-defense. In this example, Eva plays Eva offense; Esther plays defense. Esther H T T Eva and Esther each puts a penny on the table at the same time. If Esther matches Eva (both heads or both tails), then Eva pays Esther $. But if Esther fails to match Eva (one heads, one tails) Esther pays Eva $ This is called a zero-sum game, because whatever amount one player wins, the other must lose. The game has no Nash equilibrium with pure (non randomized) strategies. Offense vs Defense>Matching Pennies p 5 H EC DD & EE / Manove Clicker Question p 6
Dynamic Games So far, we ve analyzed static games, in which all players move at the same time. Now we will examine dynamic games, in which players move at different times. Dynamic Game Example: Airline fares British Airways sets its Boston-London fares. Then, Delta Airlines sets its Boston-London fares. EC DD & EE / Manove Dynamic Games p 7 The Battle of the Sexes: Static Version emember the Battle of the Sexes? wants to go to a football match, but wants to go to the opera. If they both do, then gets utility, and gets, and if they both do, then gets and gets. But if they do different things, then both get. Both must choose their strategies at the same time, without knowing what the other has done. There are two Nash equilibria:, and,. EC DD & EE / Manove Dynamic Games>Battle of the Sexes p 8
The Battle of the Sexes: Dynamic Version Now suppose that the players move at different times, first one, then the other. or example, suppose that moves first: she buys a ticket for either the football match or the opera. She shows her ticket, so he knows what she has done. Then moves: he buys his ticket for either the football match or the opera. EC DD & EE / Manove Dynamic Games>Battle of the Sexes p 9 What would happen in this game? The answer is clear! (the selfish beast ) will choose football and force to choose football as well., still looks like a Nash equilibrium. We know they won t choose,, but is, still an equilibrium? To find out, we must model strategies properly. If moves first, and see the result before he moves, then the matrix above does not correctly represent the game. EC DD & EE / Manove Dynamic Games>Battle of the Sexes p???
Dynamic-Game Strategies A strategy is a complete plan of action that specifies what a player will do in every circumstance that she can observe. rom what strategies does choose? and (as before). What about? What are his strategy choices? and are NOT strategies for. A strategy is a plan that might tell you to do different things in each situation you know about. knows has bought or bought. So his strategy must reflect his knowledge of her action. EC DD & EE / Manove Dynamic Games>Strategies p s possible strategy choices are the following (with my own nicknames): : If she bought, I will choose. If she bought, I will choose. Copy : If she bought, I will choose. If she bought, I will choose. Opposite : If she bought, I will choose. If she bought, I will choose. : If she bought, I will choose. If she bought, I will choose. These four strategies form s strategy space. EC DD & EE / Manove Dynamic Games>Strategies p
epresenting the Dynamic Game The dynamic Battle of the Sexes can be represented as follows: Copy Notice that if does, then s strategies and Copy require the same actions and lead to the same payoffs. EC DD & EE / Manove Dynamic Games>Normal orm p 3 But what are the Nash equilibria of this game? If we check each cell, we can see that there are exactly 3 pure-strategy equilibria: Copy Opposite Opposite,, Copy, In each equilibrium, the players have no incentive at the beginning of the game to deviate from their chosen strategies. However, it turns out that only, Copy is formed from strategies (plans) that would actually be followed during the game. What s wrong with the strategies in the other equilibria? Answer: They are not time-consistent EC DD & EE / Manove Dynamic Games>Equilibria p 4
Course Evaluations Now we ll do the course evaluations. The lecture will continue afterwards. The entire evaluation (except for Q4) is about M. Manove. The Ts will distribute their own evaluations. Q5: Substitute for the original question: I found the clicker questions useful. EC DD & EE / Manove Course Evaluations p 5 EC DD & EE / Manove Clicker Question p 6
Time Consistency A strategy is a plan of action that specifies what a player will do in every circumstance that she can observe. Think of a strategy as a plan made at the beginning of the game. The strategy is time-consistent if the player is willing to follow her plan as the game progresses. Example: Your strategy is to study economics tonight even if your roommate is having a party, but when the party begins, you succumb to temptation and decide not to study. Your strategy was not time-consistent. EC DD & EE / Manove Dynamic Games>Time Consistency p 7 In our Battle-Sexes example, buys her ticket first. But if says he will go to opera no matter what does, wouldn t be forced to buy an opera ticket?, is a Nash equilibrium! Copy Opposite Maybe would ignore s statement! suspects that if she chooses, will change?? his mind about. She thinks: might choose when he s planning his strategy at the beginning of the game, but when it s his turn to buy a ticket, may be unwilling to follow the - plan if I have chosen. may be an idle threat (that will not carry out), a threat that doesn t believe. EC DD & EE / Manove Dynamic Games>Time Consistency p 8
A New Kind of Equilibrium In general, the Nash equilibrium does not guarantee that equilibrium strategies will be time consistent, because the Nash-equilibrium concept doesn t eliminate idle threats. However, there s a special kind of Nash equilibrium that does guarantee time-consistent equilibrium strategies: the subgame-perfect Nash equilibrium. EC DD & EE / Manove Dynamic Games>Time Consistency p 9 Normal-orm and Extensive-orm Games So far, we ve described games with a matrix in which each row or column represents a player s strategy: the normal-form game. But to find a subgame-perfect Nash equilibrium we need a different game structure: the extensive-form game. We ll explain the extensive-form game in the next lecture, and we ll use it to find an equilibrium with time-consistent strategies. EC DD & EE / Manove Dynamic Games>Time Consistency p
EC DD & EE / Manove Clicker Question p End of ile EC DD & EE / Manove End of ile p