Identify all Pareto efficient outcomes, all pure strategy Nash equilibria and mixed-strategy Nash equilibria.
Identify which of these are stable. These games are not symmetrical so, if there are mixed strategy Nash equilibria, they may have different randomization strategies.
Please also identify the type of game: Pure coordination, Asymmetric coordination (battle-of-the-sexes), Stag Hunt (trust), Chicken/Hawk-Dove (conflict) or Prisoner’s Dilemma (cooperation). Remember to refer to Nash equilibria and Pareto efficient states by the
strategies
(e.g., (A,A)),
not
by the payoffs.
Attached is a mid-quarter review which can provide some hints on how to solve if you get stuck.
Intro to Game Theory (Part_1) Below are some game matrices. identify all Pareto efficient outcomes, all pure strategy Nash equilibria and mixed-strategy Nash equilibria. Identify which of these are stable. These games are not symmetrical so, if there are mixed strategy Nash equilibria, they may have different randomization strategies. Please also identify the type of game: Pure coordination, Asymmetric coordination (battle-of-the-sexes), Stag Hunt (trust), Chicken/Hawk-Dove (conflict) or Prisoner’s Dilemma (cooperation). Remember to refer to Nash equilibria and Pareto efficient states by the strategies (e.g., (A,A)), not by the payoffs. Attached is a mid-quarter review which can provide some hints on how to solve if you get stuck. 1. A B A 0,1 2,0 B -1,3 3,6 Pareto efficient outcomes: Pure strategy NE: Mixed-strategy NE: Yes or No If Yes, P1 plays A __% of the time. P2 plays A __% of the time. Which Nash equilibria are stable (check pure and mixed): Type of game: 2. A B A -1,3 4,1 B -3,6 2,4 Pareto efficient outcomes: Pure strategy NE: Mixed-strategy NE: Yes or No If Yes, P1 plays A __% of the time. P2 plays A __% of the time. Which Nash equilibria are stable (check pure and mixed): Type of game: 3. A B A 0,4 -1,-1 B -1,-1 2,5 Pareto efficient outcomes: Pure strategy NE: Mixed-strategy NE: Yes or No If Yes, P1 plays A __% of the time. P2 plays A __% of the time. Stable Nash (check pure and mixed): Type of game: 4. A B A 1,-1 4,0 B 2,2 3,1 Pareto efficient outcomes: Pure strategy NE: Mixed-strategy NE: Yes or No If Yes, P1 plays A __% of the time. P2 plays A __% of the time. Stable Nash (check pure and mixed): Type of game: 5. A B A 2,4 1,1 B 1,1 3,3 Pareto efficient outcomes: Pure strategy NE: Mixed-strategy NE: Yes or No If Yes, P1 plays A __% of the time. P2 plays A __% of the time. Stable Nash (check pure and mixed): Type of game: Mid-quarter Review for DA-4500-2 Game Theory for Defense Analysis Beta version 1.1 1. Finding Pareto efficient outcomes: Pareto efficient outcomes are often said to be the “good” potential outcomes of any game. I think of them more as the “not obviously bad” outcomes. PE outcomes are what are left over after you eliminate the outcomes where all of the players are worse off compared to some other outcome (or where at least one player is worse off and all the other players are the same.) To demonstrate this, I will use the following game matrix: Left Right Up 0,0 1,-10 Down -11,2 1,0 The game matrix itself does not matter to Pareto efficiency. We are just comparing outcomes. There are two methods you can use, depending on what works better for you. 1A. The table method So one thing we can do is list the strategies and payoffs: Strategies Payoffs (U,L) (0,0) (U,R) (1,-10) (D,L) (-11,2) (D,R) (1,0) I recommend going through the outcomes one-by-one. I will call the outcome that you are checking the “focal outcome.” Ask yourself: (1) Are there other outcomes where P1 is better off than in the focal outcome? (2) If yes, for any of these other outcomes, are all of the other players also better off or the same? (3) If yes, then the focal outcome is not Pareto efficient and can be eliminated. (4) Do procedures (1) – (3) for each player. (5) Do procedures (1) – (4) for each outcome. (6) Whatever is left are the Pareto efficient outcomes. Remember to report them using the strategies. For example, report “(U,L).” Do not report “(0,0)”. Using the example: Check strategy (U,L). The focal outcome for this strategy is (0,0) (1) Yes. P1 is better off at Strategy (U,R) and (D,R). (2) Yes. P2 has the same payoff at (D,R). (3) Therefore, (U,L) is not Pareto efficient since P1 does better at (D,R) and P2 does the same. It can be eliminated. Check strategy (U,R). The focal outcome for this strategy is (1,-10) (1) No. Player 1 is not better off at another outcome. (2) N/A (3) N/A (4) Now we check P2. (1) Yes. P2 is better off at (U,L) and (D,L) and (D,R). (2) Yes. P1 is the same at (D,R). (3) Therefore, (U, R) is not Pareto efficient since P2 does better at (D,R) and P1 does the same. It can be eliminated. Check strategy (D, L). The focal outcome for this strategy is (-11, 2) (1) Yes. Player 1 is better off at (U, L), (U, R) and (D, L). (2) No. Player 2 is not better off at any of these states. (3) N/A (4) Now we check P2. (1) No. P2 is not better at any other outcome. (2) N/A (3) N/A (4) We have checked all of the players and not eliminated (U, R). Therefore (U, R) is a Pareto efficient outcome. Note: if there is only one outcome that is best for any of the players (no ties) it is automatically Pareto efficient. Check strategy (D, R). The focal outcome for this strategy is (1, 0) (1) No. Player 1 is not better off at any other outcome. (2) N/A (3) N/A (4) Now we check P2. (1) Yes. P2 is better off at (D, L). (2) No. P1 is worse off at (D, L). (3) N/A (4) We have checked all of the players and not eliminated (D, L). Therefore (D, L) is a Pareto efficient outcome. Note: if there is only one outcome that is best for any of the players (no ties) it is automatically Pareto efficient. This game matrix has two Pareto efficient outcomes (D, L) and (U, R). 1B. The graphic method Some people find it more helpful to look at a graph than at a table. You can graph this out and follow the same procedure as for the table method. Here it might be more easy to see that (D, R) is better than (U, L) and (U, R), but (D, L) is better for Player 2 than (D, R). Therefore, (D, R) and (D, L) are Pareto efficient outcomes (they are the Pareto frontier). 2. Finding Pure Strategy Nash Equilibria: A pure strategy Nash equilibrium occurs when players do not randomly mix strategies and they do not have an incentive to switch to another strategy. To demonstrate how to find pure strategy Nash equilibria, we can use the following procedure: For each strategy set check: (1) Does P1 have an incentive to switch strategies (all other players play the same strategy)? (2) If so, this is not a pure strategy Nash equilibrium. (3) Repeat (1) and (2) for each player. (4) If no player has an incentive to switch strategies unilaterally, this is a pure strategy Nash equilibrium. As an example, we will use this game matrix: Left Right Up 5,3 2,2 Down 4,0 3,1 Check strategy (U, L). (1) If P1 switches to Down, P1 receives a lower score, so P1 has no incentive to switch. (2) N/A (3) Now we’ll check P2 (1) If P2 switches to Right, P2 receives a lower score, so P2 has no incentive to switch. (2) N/A (4) Since neither player has an incentive to switch (U, L) is a pure strategy Nash equilibrium. Check strategy (U, R). (1) If P1 switches to Down, P1 receives a higher score, so P1 has an incentive to switch. (2) Therefore, (U, R) is not a pure strategy Nash equilibrium. Check strategy (D, L). (1) If P1 switches to UP, P1 receives a higher score, so P1 has an incentive to switch. (2) Therefore (D, L) is not a pure strategy Nash equilibrium. Check strategy (D, R). (1) If P1 switches to Up, P1 receives a lower score, so P1 has no incentive to switch. (2) N/A (3) Now we’ll check P2 (1) If P2 switches to Left, P2 receives a lower score, so P2 has no incentive to switch. (2) N/A (4) Since neither player has an incentive to switch (D, R) is a pure strategy Nash equilibrium. Therefore, the pure strategy Nash equilibria for this game are (U, L) and (D, R). 3. Finding Mixed-Strategy Nash equilibria: A mixed-strategy Nash equilibrium occurs when a player randomly chooses between two or more strategies. This often occurs in strategic contests where a player needs to be unpredictable. The goal is to make it so that the player cannot be exploited by the other player in the long run. 3A. An example (where there is a mixed-strategy NE) We will look the game matrix below: Left Right Up 5,3 2,2 Down 4,-1 3,1 The procedure is as follows: (1) P1 wants to figure out how to randomize so that P2 cannot get an advantage. Therefore wants to make P2’s expected payoffs equal for either choice that P2 makes. P1 wants to find the fraction of time that P1 should play a strategy to make P2’s expected payoffs equal. P1 uses the variable to represent the fraction of the time P1 will choose “up.” P1 is trying to solve for. (2) Calculate the expected payoffs to P2 if P2 plays “Left” in terms of. We have to use the payoff to P2 if P1 plays “Up” and multiply that by the fraction of the time that P1 plays “Up.” We also have to use the payoff to P2 if P1 plays “Down” and multiply it by the faction of the time that P2 plays “Down,”. (2) Calculate the expected payoffs to P2 if P2 plays “Right” in terms of. We, again, use the payoff to P2 if P1 plays “Up” and multiply that by the fraction of the time that P1 plays “Up.” We also have to use the payoff to P2 if P1 plays “Down” and multiply it by the faction of the time that P2 plays “Down,”. (3) Now we set these equal and help P1 solve for. Therefore, at a mixed strategy Nash equilibrium, P1 will randomize and play “Up” 2/3 of the time and “Down” 1/3 of the time. Note that if you find that or, this