ECO 5341 Strategic Behavior Lecture Notes 3

Transcription

1 ECO 5341 Strategic Behavior Lecture Notes 3 Saltuk Ozerturk SMU Spring 2016 (SMU) Lecture Notes 3 Spring / 20

2 Lecture Outline Review: Dominance and Iterated Elimination of Strictly Dominated Strategies (Pure-Strategy) Nash Equilibrium How to Find Pure-Strategy Nash Equilibrium (SMU) Lecture Notes 3 Spring / 20

3 Games of Chicken Games of chicken comes from a game that was supposedly played by American teenagers in the 1950s: 1 Two teenagers take their cars to opposite ends of Main Street at mid-night and start to drive toward each other. 2 The one who swerve to prevent a collision is the chicken, and the one who keeps going straight is the winner. 3 If both maintain a straight course, there is a collision where both cars are damaged and both players injured. Bob Charles Swerve (Chicken) Straight (Tough) Swerve (Chicken) 0, 0 1, 3 Straight (Tough) 3, 1 5, 5 Games of chicken illustrates a di erent kind of coordination: the players want to avoid, not choose, actions with the same labels and coordination failure may result in di erent outcomes. (SMU) Lecture Notes 3 Spring / 20

4 Iterated Elimination of Strictly Dominated Strategies Review Consider a price competition game between Toys R Us and Wal-Mart where each of them has to decide (independently) whether to sell a particular toy at a High or Low price. High Low High 10, 10 2, 15 Low 15, 2 5, 5 Remark: For static games, decisions about eliminating dominated strategies are made as thought experiments, not through rounds of maneuvers. (SMU) Lecture Notes 3 Spring / 20

5 Strictly Dominant Strategies Strictly Dominant Strategies Suppose a i and b i are two strategies of player i. Strategy a i strictly dominates strategy b i if a i always does strictly better than b i, no matter what the rivals moves are, i.e., a i strictly dominates b i if u i (a i, s i ) > u i (b i, s i ), for all s i = (s 1, s 2,..., s i 1, s i +1,..., s n ) Strategy a i is a strictly dominant strategy for player i if a i strictly dominates every other possible strategy of player i. If player i is rational and has a strictly dominant strategy a i, then it would be reasonable that player i will not play any other strategy. (SMU) Lecture Notes 3 Spring / 20

6 Weakly Dominant Strategies Weakly Dominant Strategies Suppose a i and b i are two strategies of player i. Strategy a i weakly dominates strategy b i if, no matter what the rivals moves are, playing a i is at least as good as playing b i, and there are some situations that playing a i is strictly better than playing b i for player i. I.e., a i weakly dominates b i if u i (a i, s i ) u i (b i, s i ), for all s i and u i (a i, s i ) > u i (b i, s i ), for some s i. Strategy a i is a weakly dominant strategy for player i if a i weakly dominates every other possible strategy of player i. If player i has a weakly dominant strategy and player i is rational and cautious, then she will play the weakly dominant strategy. (SMU) Lecture Notes 3 Spring / 20

7 An Example Three players, Alice, Betty and Cathy, all live on the same small street. Each is asked to contribute toward the creation of a ower garden at the intersection of their small street with the main highway. Each has two choices, contribute or not contribute. The three players make choices simultaneously. Game table: Alice Betty C NC C 5, 5, 5 3, 6, 3 NC 6, 3, 3 4, 4, 1 C Alice Cathy Betty C NC C 3, 3, 6 1, 4, 4 NC 4, 1, 4 2, 2, 2 NC Questions: strictly dominant (dominated) strategies? dominance solvable? can one have more than one strictly dominant strategy in a game? (SMU) Lecture Notes 3 Spring / 20

8 Solving Static Games Nash Equilibrium Not all games are dominance solvable. More general solution concepts? The solution concept that always exists for all nite games is Nash equilibrium. (SMU) Lecture Notes 3 Spring / 20

9 John Forbes Nash, Jr John Forbes Nash, Jr. (born June 13, 1928), is an American mathematician who works in game theory, di erential geometry, and partial di erential equations, serving as a Senior Research Mathematician at Princeton University. He shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi. (Wikipedia) (SMU) Lecture Notes 3 Spring / 20

10 Nash Equilibrium De nition A strategy s i is a best response for player i to the opponents strategies s i if there is no strategy for player i that leads to higher payo s when the opponents play s i. For a two-player simultaneous-move game, a Nash equilibrium is a pair of strategies that are best responses to one another. A Nash equilibrium is a stable situation/outcome that no player would like to deviate (unilaterally) if other players stick to the situation/outcome. (SMU) Lecture Notes 3 Spring / 20

11 Nash Equilibrium An Illustrative Example Consider the following abstract two-player simultaneous game: L C R T 0, 4 4, 0 3, 3 M 4, 0 0, 4 3, 3 B 3, 3 3, 3 3.5, 3.6 The combination of strategies (B, R) a strategy pro le features: The row player cannot do better by choosing a strategy di erent from B, given that the column player is playing R; The column player cannot do better by choosing a strategy di erent from R, given that the row player is playing B. No player has any incentive to deviate from (B, R), given the other player s strategy. (B, R) is called a Nash equilibrium. (SMU) Lecture Notes 3 Spring / 20

12 Nash Equilibrium Examples Battle of the Sexes: Prisoner s Dilemma: Pure Coordination Game: Gambling Opera Gambling 1, 2 0, 0 Opera 0, 0 2, 1 Stay Silent Confess Stay Silent -2, -2-5, 0 Confess 0, -5-4, -4 Left Right Left 7, 7 0, 0 Right 0, 0 7, 7 (SMU) Lecture Notes 3 Spring / 20

13 How to Find Pure-Strategy Nash Equilibria Cell-by-Cell Inspection Cell-by-Cell Inspection is a low-tech way of nding pure-strategy Nash equilibria. But this method always works for ( nite) static games. Cell-by-Cell Inspection : Check each cell in the bimatrix (game table) to see if either side has a pro table deviation. A pro table deviation is where by changing his strategy (leaving the rival s choice xed) a player can improve his or her payo s. If there is no pro table deviation for anyone, the cell corresponds to the result of a pair of best responses. Look for all pairs of best responses. Cell-by-Cell Inspection method nds all Pure-Strategy Nash Equilibria for a given game table. However, this method is time-consuming for more complicated games and sometimes this method does not work for pure NE (why?). (SMU) Lecture Notes 3 Spring / 20

14 How to Find Pure-Strategy Nash Equilibria A Two-Player Bargaining Game Consider a two-player bargaining game (negotiation game): 1 Suppose you are in a negotiation with another party over the allocation of certain resources. Each of you makes demands regarding the size of the pie. 2 If the sum of the two demands is less than or equal to the size of the pie, then each of the two players gets what she demands; 3 If the sum of the two demands is larger than the size of the pie, then there is an impasse and no one gets anything; 4 Furthermore, the size of the pie is 100 and each player can demand Low (25), Medium (50), or High (75). (SMU) Lecture Notes 3 Spring / 20

15 How to Find Pure-Strategy Nash Equilibria A Two-Player Bargaining Game Bimatrix of the two-player bargaining game: (SMU) Lecture Notes 3 Spring / 20

16 How to Find Pure-Strategy Nash Equilibria A Two-Player Bargaining Game Bimatrix of the two-player bargaining game: Pure-Strategy Nash Equilibria: (SMU) Lecture Notes 3 Spring / 20

17 Cell-By-Cell Inspection Cannot Find All Nash Equilibria Cell-By-Cell Inspection method only nds pure-strategy Nash equilibrium. This method may not nd any pure-strategy Nash equilibrium for some games. Example Matching Pennies: Another Example: Head Tail Head 1, -1-1, 1 Tail -1, 1 1, -1 Employees can work or shirk (salary: 100K, cost of e ort: $50K) Managers can monitor or not (payo $200K if employee works, $0 if employee shirks, cost of monitoring: $10K) Monitor Not Monitor Work 50, 90 50, 100 Shirk 0, , -100 (SMU) Lecture Notes 3 Spring / 20

18 More on the concept of Nash Equilibrium The concept of Nash equilibrium was introduced by John Nash at the age of 21, while he was still a math graduate student at Princeton. John Nash in his 1950 paper Equilibrium points in N-Person Games, introduced the concept of Nash equilibrium and proved that for any ( nite) N-player normal-form games, there exists at least one (pure-strategy or mixed-strategy) Nash equilibrium. John Nash, together with Reinhard Selten and John C. Harsanyi, received the Nobel Prize in economics in Main contributions of his 1950 paper Equilibrium points in N-Person Games: Nash equilibrium concept: this concept has spawned much of the literature on non-cooperative game theory, which has totally changed almost every subject in economics; Proof of existence: John Nash used Kakutani s xed-point theorem to prove the existence of Nash equilibrium. This method of proof was later employed by economists to show existence of equilibrium in various subjects of economics. (SMU) Lecture Notes 3 Spring / 20

19 Mathematical De nition of Nash Equilibrium In an n-player normal-form game fs 1, S 2,..., S n ; u 1, u 2,..., u n g, a strategy pro le (s1, s 2,..., s n ) is a Nash equilibrium if, for every player i, si is (at least tied for) player i s best response to the strategies speci ed for the (n 1) other players, s1,..., s i 1, s i+1,..., s n. In other words, u i (s 1,..., s i 1, s i, s i+1,..., s n ) u i (s 1,..., s i 1, s i, s i+1,..., s n ) for every feasible strategy s i of player i (for every s i 2 S i ). Question: how to relate the above to our previous de nition a Nash equilibrium is a stable situation where no player has any incentive to deviate, given the other players strategies? (SMU) Lecture Notes 3 Spring / 20

20 Nash Equilibrium An Application Child s Play Wins Auction House an Art Sale, NYTimes, 04/2005. A Japanese art collector intended to choose Christie s or Sotheby s (auction houses) to handle the sale of his $20 million collection. He was reluctant to split the collection between the two houses and he opted for having the auction houses play Rock, Paper, Scissors. The two houses had a week to come up with a strategy and winning the game meant millions of dollars in commission. Sotheby s didn t pay too much attention (just a game of chance). Christie s did it di erently the president of Christie s in Japan spent a week to do extensive research. The most useful advice came from a sta s twins (11) who play the game constantly in school: Start with scissors because rock is too obvious and scissors beats paper, and when there is a tie in the rst game, stick to scissors because everyone expects you to switch to rock. Christie s won in the rst round (Sotheby s chose paper)... (SMU) Lecture Notes 3 Spring / 20

21 Final Remarks on Applying NE Concept Although NE exists for all games, NE outcomes should at best be used as guidelines instead of ultimate solutions. One common issue regarding NE is multiple Nash equilibria. Examples of Multiple NE s: Chicken Tough Chicken 0,0-1,3 Tough 3,-1-5,-5 A B C X A 5,5 0,0 0,0 0,0 B 0,0 5,5 0,0 0,0 C 0,0 0,0 5,5 0,0 X 0,0 0,0 0,0 5,5 Focal points (Schelling (1960)); E orts in avoiding coordination failures (e.g., pre-election polls and sometimes cheap talk); Equilibrium selection: L R T 1,1 0,0 B 0,0 0,0 (SMU) Lecture Notes 3 Spring / 20