3 2 1 Static Games of complete information

3. 2. 1. Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies Strategic Behavior in Business and Econ

Outline 3. 1. What is a Game ? 3. 1. 1. The elements of a Game 3. 1. 2 The Rules of the Game: Example 3. 1. 3. Examples of Game Situations 3. 1. 4. Types of Games 3. 2. Solution Concepts 3. 2. 1. Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies 3. 3. 2. Dynamic Games of complete information: Backward Induction and Subgame perfection Strategic Behavior in Business and Econ

Reminder Static Games of Complete Information All the players choose their strategies simultaneously. This does not mean “at the same time” but “without knowing the choice of others” Because of this simultaneity they can be represented by means of a table (payoff matrix) They are “one-shot games”, that is, they are played only once All the players have all the information regarding who are the other players, what are the own strategies and the strategies of the others, what are the own payoffs and the payoffs of the others, and what are the rules of the game Strategic Behavior in Business and Econ

Reminder Solution concepts for this type of games Equilibrium in Dominant Strategies When there is an “always winning” strategy Equilibrium by elimination of Dominated Strategies When there are “worse than” strategies Nash Equilibrium Works in any case In pure strategies (players do not randomize) In mixed strategies (players do randomize) Strategic Behavior in Business and Econ

Reminder Nash Equilibrium A Nash Equilibrium is a combination of strategies by the players with the special feature that: All players are playing a best reply to what the other players are doing Notice that, since all the players are playing a “best reply”, nobody will want to change his choice of an strategy !!! Is in this sense that a Nash Equilibrium is stable Strategic Behavior in Business and Econ

Reminder Nash Equilibrium In practical terms A Nash Equilibrium is where the best replies of the players coincide That is, a Nash Equilibrium is where the red circles coincide (in the Table representation of the game) Strategic Behavior in Business and Econ

Example: The Rock-Paper-Scissors Game And there is no coincidence of red circles. !! There is no stable outcome ! Player 2 Rock Player 1 Rock Paper Scissors 0, 0 +1, -1 -1, +1 Paper -1, +1 0, 0 +1, -1 Strategic Behavior in Business and Econ Scissors +1, -1 -1, +1 0, 0

Example: The Rock-Paper-Scissors Game In this games, players choose their strategy “at random” This will be the equilibrium Player 2 Rock Player 1 Rock Paper Scissors 0, 0 +1, -1 -1, +1 Paper -1, +1 0, 0 +1, -1 Strategic Behavior in Business and Econ Scissors +1, -1 -1, +1 0, 0

How does people play this game ? It is well known that people plays the game “at random” in such a way that each strategy is chosen with the same probability (1/3). Suppose that Player 2 plays at random. What will be the Expected Payoff for Player 1 at each of his/her strategies ? E(Rock) = 1/3 · 0 + 1/3 · (- 1) + 1/3 · (+ 1) = 0 E(Paper) = 1/3 · (+ 1) + 1/3 · 0 + 1/3 · (- 1) = 0 E(Scissors) = 1/3 · (- 1) + 1/3 · (+ 1) + 1/3 · 0 = 0 Thus, ANY strategy is a “best reply” to Player 2 playing “at random” (all of them produce the same payoff, namely 0) Strategic Behavior in Business and Econ

Therefore, if Player 2 plays each strategy with equal probability (1/3), Player 1 can play in any way: Play Rock Play Paper Play Scissors Or … Randomize ! (The payoff will be 0 always) In other words, Player 2 is playing in such a way that he/she Makes Player 1 completely indifferent among his/her strategies. Player 1 doesn't a have a “clear” strategy to use, anything is as good as anything else ! Strategic Behavior in Business and Econ

Imagine now that Player 2 plays at random but with different probabilities: p(Rock) = ¼ p(Paper) = ¼ p(Scissors) = ½ What will be now the Expected Payoff for Player 1 at each of his/her strategies ? E(Rock) = ¼ · 0 + ¼ · (- 1) + ½ · (+ 1) = ¼ E(Paper) = ¼ · (+ 1) + ¼ · 0 + ½ · (- 1) = - ¼ E(Scissors) = ¼ · (- 1) + ¼ · (+ 1) + ½ · 0 = 0 Strategic Behavior in Business and Econ

Imagine now that Player 2 plays at random but with different probabilities: p(Rock) = ¼ p(Paper) = ¼ p(Scissors) = ½ What will be now the Expected Payoff for Player 1 at each of his/her strategies ? E(Rock) = ¼ · 0 + ¼ · (- 1) + ½ · (+ 1) = ¼ E(Paper) = ¼ · (+ 1) + ¼ · 0 + ½ · (- 1) = - ¼ E(Scissors) = ¼ · (- 1) + ¼ · (+ 1) + ½ · 0 = 0 Strategic Behavior in Business and Econ

Therefore, if Player 2 plays p(Rock) = ¼ p(Paper) = ¼ p(Scissors) = ½ The “best reply” for Player 1 will be to play “Rock” But if Player 1 plays “Rock”, the “best reply” for Player 2 will be to play “Paper” But if Player 2 plays “Paper”, the “best reply” for Player 1 will be to play “Scissors” But. . . No equilibrium !!! Strategic Behavior in Business and Econ

Nash Equilibrium in mixed strategies There are games in which there is no coincidence of best replies. That is, apparently, the Nash Equilibrium does not seem to exist. We will see that in such cases, because of the own nature of the game, it does not make sense for the players to play always the same strategy In such cases, players want to play “at random” We will see that such random playing (mixed strategies) corresponds to the idea of Nash Equilibrium, that is, given what the other players are doing, playing at random is my best reply Strategic Behavior in Business and Econ

Example: Soccer Penalty Kicks In soccer, a “penalty kick” is a situation of the game in which a field player (the kicker) has a “free” shoot at goal with only the goalkeeper (the goalie) to stop the shot Strategic Behavior in Business and Econ

The environment of the game Players: Strategies: Kicker and Goalie Payoffs: +1 for the Kicker if scores, -1 if fails +1 for the Goalie if blocks, -1 if fails Kicker: Shoot to the right of the goal Shoot to the left of the goal Goalie: Jump to the right of the goal Jump to the left of the goal The Rules of the Game Timing of moves Nature of conflict and interaction Information conditions Simultaneous Conflict Symmetric Strategic Behavior in Business and Econ

In real games, there are more options (shoot up, shoot down, shoot hard, shoot soft, shoot at the center, etc) We will assume, for simplicity, that when the Goalie jumps to the same side where the Kicker shot, then he blocks the shot Otherwise, the Kicker scores. Strategic Behavior in Business and Econ

Example: Soccer Penalty Kicks Goalie Left Right Left -1 , 1 1 , -1 Right 1 , -1 -1 , 1 Kicker Strategic Behavior in Business and Econ

Example: Soccer Penalty Kicks Look for the Best Replies Goalie Left Right Left -1 , 1 1 , -1 Right 1 , -1 -1 , 1 Kicker Strategic Behavior in Business and Econ

Example: Soccer Penalty Kicks There is no coincidence of best replies Apparently, no equilibria How would you play this game? What would you do if you know that the goalie jumps left 75% of the time ? Strategic Behavior in Business and Econ

Example: Soccer Penalty Kicks First guess: If the Goalie jumps left 75% of the times, I should shoot right 75% of the times Remember (probability properties): p(A and B) = p(A) x (B) p(kick left and jump left) = p(kick left) x p(jump left) Blocks p(kick left and jump right) = p(kick left) x p(jump right) Goal ! p(kick right and jump left) = p(kick right) x p(jump left) Goal ! p(kick left and jump left) = p(kick left) x p(jump left) Blocks Strategic Behavior in Business and Econ

Example: Soccer Penalty Kicks First guess: If the Goalie jumps left 75% of the times, I should shoot right 75% of the times Remember (probability properties): p(A or B) = p(A) + (B) p(scoring) = p(L and R) + p(R and L) = 0. 5625 + 0. 0625 = 0. 625 Is this all the best the Kicker can do ? Strategic Behavior in Business and Econ

Example: Soccer Penalty Kicks Second guess: If the Goalie jumps left 75% of the times, I should shoot right all the time ! (100%) p(scoring) = P(L and R) + p(R and L) = 0. 75 + 0 = 0. 75 This is the best the Kicker can do !! Strategic Behavior in Business and Econ

Example: Soccer Penalty Kicks “Real” goalies and kickers do randomize !! Suppose that the goalie jumps left with probability p (and right 1 -p) What is the kicker’s best reply? If p=1, goalie always jumps left I should kick right If p=0, goalie always jumps right I should kick left If p=? , goalie “mixes” strategies I should compute some “expected value” Strategic Behavior in Business and Econ

Example: Soccer Penalty Kicks If p=? , goalie “mixes” strategies I should compute some “expected value” E(Kick left) = p·(-1)+(1 -p)·(+1) = 1 – 2 p E(Kick right)= p·(+1)+(1 -p)·(-1) = 2 p – 1 Which one is better ? Left if 1 – 2 p > 2 p – 1, that is, p < ½ Right if 1 – 2 p < 2 p – 1, that is, p > ½ “Any” if 1 – 2 p = 2 p – 1, that is, p = ½ Strategic Behavior in Business and Econ

Example: Soccer Penalty Kicks If p=? , goalie “mixes” strategies I should compute some “expected value” E(Kick left) = p·(-1)+(1 -p)·(+1) = 1 – 2 p E(Kick right)= p·(+1)+(1 -p)·(-1) = 2 p – 1 Which one is better ? Left if 1 – 2 p > 2 p – 1, that is, p < ½ Right if 1 – 2 p < 2 p – 1, that is, p > ½ “Any” if 1 – 2 p = 2 p – 1, that is, p = ½ Strategic Behavior in Business and Econ This is the Best Reply of the Kicker

Example: Soccer Penalty Kicks Knowing the Best Reply of the Kicker, what it the best strategy for the Goalie ? Goalie’s p L (p = 1) R (p = 0) p = 0. 75 p = 0. 25 p = 0. 50 Kicker’s strategy R L R Any Strategic Behavior in Business and Econ Goalie’s Payoff -1. 0 -0. 5 -0. 1 0

Example: Soccer Penalty Kicks Knowing the Best Reply of the Kicker, what it the best strategy for the Goalie ? Goalie’s p L (p = 1) R (p = 0) p = 0. 75 p = 0. 25 p = 0. 50 Kicker’s strategy R L R Any Strategic Behavior in Business and Econ Goalie’s Payoff -1. 0 -0. 5 -0. 1 0

Example: Soccer Penalty Kicks Knowing the Best Reply of the Kicker, what it the best strategy for the Goalie ? The best strategy for the Goalie is to jump left or right with the same (p = 0. 5) probability • • This is NOT the equilibrium of the game. This is only to show that the best strategy for the Goalie is to randomize in such a way (p = 0. 5) that the Kicker doesn't know what to do (kicking left or right yields the same expected payoff) This is always the case in games whose equilibria consists of mixed (random) strategies Strategic Behavior in Business and Econ

Mixed strategies If my opponent knows what I will do, I will always lose ! Randomizing just right takes away any ability to be taken advantage of Make opponent indifferent between her strategies Implications (strangely) A player chooses his strategy so as to make her opponent indifferent If done right, the other player earns the same payoff from either of her strategies Use the strategy that keeps your opponent guessing !! Strategic Behavior in Business and Econ

Example: A (simple) Principal-Agent game Manager Employee Monitor No Monitor Work 50 , 90 50 , 100 Shirk 0 , -10 100 , -100 Notice that there is no equilibrium !! Strategic Behavior in Business and Econ

Example: A (simple) Principal-Agent game Manager Monitor (q) Employee No Monitor (1 -q) Work (p) 50 , 90 50 , 100 Shirk (1 -p) 0 , -10 100 , -100 But there is always an equilibrium ! Employee chooses (work, shirk) with probabilities (p, 1 -p) Manager chooses (monitor, no monitor) with probabilities (q, 1 -q) Strategic Behavior in Business and Econ

Example: A (simple) Principal-Agent game The key to find the equilibrium in mixed strategies is always the same Find the probability (p) that player 1 must use as to make player 2 indifferent between her strategies Find the probability (q) that player 2 must use as to make player 1 indifferent between her strategies • Strategic Behavior in Business and Econ

Example: A (simple) Principal-Agent game Manager Monitor (q) Employee No Monitor (1 -q) Work (p) 50 , 90 50 , 100 Shirk (1 -p) 0 , -10 100 , -100 In this case: 1. Find p so that 2. Find q so that E(Monitor) = E(No Monitor) E(Work) = E(Shirk) Strategic Behavior in Business and Econ

Example: A (simple) Principal-Agent game Manager Monitor (q) Employee No Monitor (1 -q) Work (p) 50 , 90 50 , 100 Shirk (1 -p) 0 , -10 100 , -100 1. Find p so that E(Monitor) = E(No Monitor) p·(90) + (1 -p)·(-10) = p·(100) + (1 -p)·(-100) -10 + 100 p = -100 +200 p 90 = 100 p p = 9/10 = 0. 9 (90%) Strategic Behavior in Business and Econ

Example: A (simple) Principal-Agent game Manager Monitor (q) Employee No Monitor (1 -q) Work (p) 50 , 90 50 , 100 Shirk (1 -p) 0 , -10 100 , -100 2. Find q so that E(Work) = E(Shirk) q·(50) + (1 -q)·(50) = q·(0) + (1 -q)·(100) 50 = 100 - 100 q = 50 q = 1/2 = 0. 5 (50%) Strategic Behavior in Business and Econ

Example: A (simple) Principal-Agent game Manager Monitor (q) Employee No Monitor (1 -q) Work (p) 50 , 90 50 , 100 Shirk (1 -p) 0 , -10 100 , -100 Thus we have p = 9/10 = 0. 9 (90%) The Worker should work 90% of the time (and shirk 10%) The Manager should monitor 50% of the time Strategic Behavior in Business and Econ q=1

Example: A (simple) Principal-Agent game Notice: The worker's strategy was found using the manager's payoffs The manager's strategy was found using the worker's payoffs The idea is to choose one player's strategy so that it makes the other player indifferent between her strategies Strategic Behavior in Business and Econ

Example: The Battle of the Sexes Chris There are two Nash Equilibria Football Opera 3, 1 0, 0 -1 , -1 1, 3 Pat Football Strategic Behavior in Business and Econ

Example: The Battle of the Sexes But there is another Equilibrium in Mixed Strategies Chris Opera (q) Football (1 -q) Opera (p) 3, 1 0, 0 Football (1 -p) -1 , -1 1, 3 Pat Strategic Behavior in Business and Econ

Example: The Battle of the Sexes Chris Opera (p) Opera (q) Football (1 -q) 3, 1 0, 0 Pat Football (1 -p) 1. Find p so that -1 , -1 1, 3 E(Opera) = E(Football) (for Player 2) p·(1) + (1 -p)·(-1) = p·(0) + (1 -p)·(3) 2 P - 1 = 3 - 3 p 5 p = 4 1. p = 4/5 = 0. 8 (80%) Strategic Behavior in Business and Econ

Example: The Battle of the Sexes Chris Football (1 -q) Opera (p) 3, 1 0, 0 Pat Football (1 -p) 2. Find q so that -1 , -1 1, 3 E(Opera) = E(Football) (for Player 1) q·(3) + (1 -q)·(0) = q·(-1) + (1 -q)·(1) 3 q = 1 - 2 q 5 q = 1/5 = 0. 2 (20%) Strategic Behavior in Business and Econ

Example: The Battle of the Sexes Chris Football (1 -q) Opera (p) 3, 1 0, 0 Pat Football (1 -p) -1 , -1 1, 3 Thus we have p = 4/5 = 0. 80 (80%) q = 1/5 = 0. 2 (20%) Pat should go to the Opera 80% of the time (and 20% Football) Chris should go to the Football 20% of the time (and 80% Opera) Strategic Behavior in Business and Econ

Summary A mixed strategy is a way of playing the game by choosing each strategy available with some probability Mixed strategies are computed looking at the other player's payoffs The idea is to choose one player's strategy so that it makes the other player indifferent between her strategies Randomizing just right takes away any ability to be taken advantage of In any game, there is always an equilibrium, either on pure or in mixed strategies Strategic Behavior in Business and Econ

Final word: What does it mean to play randomly ? Use the mixed strategy that keeps your opponents guessing BUT your probability of each action must be the same every time Example: In the Worker-Manager game, the Manager should not Monitor every other day !! The Manager should flip a coin everyday to decide if she monitors or not Example: When playing the Rock-Paper-Scissors game the equilibrium mixed strategy is to play each strategy with the same probability (1/3), but you don't want to play a sequence: R-P-S-R-P-S- · · · Humans are very bad at this. Exploit patterns !!! (and don't follow a pattern !!!) Strategic Behavior in Business and Econ