Strategic Behavior in Business and Econ 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 ? The elements of a Game The Rules of the Game: Example Examples of Game Situations Types of Games 3.2. Solution Concepts Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies Dynamic Games of complete information: Backward Induction and Subgame perfection

Strategic Behavior in Business and Econ There are, basically, four different types of games All games in a given category are represented and solved alike Reminder

Strategic Behavior in Business and Econ Solution Concepts A solution of a game is called an Equilibrium of the game Reminder

Strategic Behavior in Business and Econ 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 Reminder

Strategic Behavior in Business and Econ 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 An equilibrium of the game is a choice of strategies by all the players that is stable, in the sense that Given what the other players are doing, nobody has any reason to change his or her own strategy

Strategic Behavior in Business and Econ Advertise Not advertise Reynolds (player 1) Philip Morris (player 2) 50, 50 Advertise Not advertise 20, 60 30, 3060, 20 Example: Game with “always winning” strategy (Dominant Strategy) Prediction of Game Theory: Both have a clear best strategy Advertise no matter what

Strategic Behavior in Business and Econ (in thousands of dollars) $2$4$5 Bar 1 $210, 1014, 1214, 15 $4 12, 1420, 2028, 15 $5 15, 1415, 2825, 25 Bar 2 Prediction of Game Theory: There is no “always winning” strategy Example: Game with “worse than” strategies (Dominated Strategies)

Strategic Behavior in Business and Econ (in thousands of dollars) $2$4$5 Bar 1 $210, 1014, 1214, 15 $4 12, 1420, 2028, 15 $5 15, 1415, 2825, 25 Bar 2 Prediction of Game Theory: But there is a clearly bad strategy: $2 is always worse than $4

Strategic Behavior in Business and Econ (in thousands of dollars) $2$4$5 Bar 1 $210, 1014, 1214, 15 $4 12, 1420, 2028, 15 $5 15, 1415, 2825, 25 Bar 2 Prediction of Game Theory: If $2 is removed from the game (it will never be used) then the game is more clear

Strategic Behavior in Business and Econ (in thousands of dollars) $4$5 Bar 1 $4 20, 2028, 15 $5 15, 2825, 25 Bar 2 Prediction of Game Theory: Now $4 is clearly the best strategy no matter what my competitor does

Strategic Behavior in Business and Econ (in thousands of dollars) $4$5 Bar 1 $4 20, 2028, 15 $5 15, 2825, 25 Bar 2 Prediction of Game Theory: NOTICE: The “coincidence” of red circles is (again) the stable outcome

Strategic Behavior in Business and Econ Mary RightLeft Paul Right 0, 0 -50, -50 Left -50, -500, 0 There are 2 equilibrium: (coincidence of red circles) Both players driving on the right Both players driving on the left (but players do not randomize!) Example: Game with no “always winning strategy with no “worse than” strategies (Nash Equilibrium) Prediction of Game Theory: There is no “always winning” nor “worse than” strategies

Strategic Behavior in Business and Econ RockPaperScissors Player 1 Rock0, 0-1, +1+1, -1 Paper +1, -10, 0-1, +1 Scissors -1, +1+1, -10, 0 Player 2 Prediction of Game Theory: There is no “always winning” nor “worse than” strategies Example: Game with no “always winning strategy with no “worse than” strategies (Nash Equilibrium)

Strategic Behavior in Business and Econ RockPaperScissors Player 1 Rock0, 0-1, +1+1, -1 Paper +1, -10, 0-1, +1 Scissors -1, +1+1, -10, 0 Player 2 Prediction of Game Theory: There is no “coincidence” of red circles Example: Game with no “always winning strategy with no “worse than” strategies (Nash Equilibrium) Players do randomize to play this game

Strategic Behavior in Business and Econ Equilibrium in Dominant Strategies A player has a Dominant Strategy if, regardless the strategy chosen by the other players, that strategy is always a best response (it has all red circles) If every player has a Dominant Strategy, then the predicted outcome of the game is the one that corresponds to the players choosing that strategy If some players have a Dominant Strategy and others don't, the predicted outcome is that players with dominant strategies will use them whereas players with no dominant strategies will choose a best response to them

Strategic Behavior in Business and Econ Advertise Not advertise Reynolds (player 1) Philip Morris (player 2) 50, 50 Advertise Not advertise 20, 60 30, 3060, 20 Example: Advertise Not advertise Reynolds (player 1) Philip Morris (player 2) 50, 80 Advertise Not advertise 20, 60 30, 3060, 20 Example:

Strategic Behavior in Business and Econ Equilibrium by elimination of Dominated Strategies A player has a Dominated Strategy if, regardless the strategy chosen by the other players, that strategy is always worse than some other strategy (it has NO red circles) If a player has a Dominated Strategy, the corresponding row or column can be removed from the table After the removal of one dominated strategy it might happen that other strategies are also dominated. The process of elimination of dominated strategies continues until there are no more dominated strategies for any player

Strategic Behavior in Business and Econ Example VWXYZ A 4,-13,0-3,1-1,4-2,0 B -1,12,22,3-1,02,5 C 2,1-1,-10,44,-10,2 D 1,6-3,0-1,41,1-1,4 E 0,01,4-3,1-2,3-1,-1 Player 2 Player 1

Strategic Behavior in Business and Econ Example VWXYZ A 4,-13,0-3,1-1,4-2,0 B -1,12,22,3-1,02,5 C 2,1-1,-10,44,-10,2 D 1,6-3,0-1,41,1-1,4 E 0,01,4-3,1-2,3-1,-1 Player 2 Player 1 Look for the best replies

Strategic Behavior in Business and Econ Example VWXYZ A 4,-13,0-3,1-1,4-2,0 B -1,12,22,3-1,02,5 C 2,1-1,-10,44,-10,2 D 1,6-3,0-1,41,1-1,4 E 0,01,4-3,1-2,3-1,-1 Player 2 Player 1 Look for the best replies Notice that (B,Z) is the only outcome with coincidence of red circles There are no strategies with “all red circles” That is, there are no Dominant Strategies

Strategic Behavior in Business and Econ Example VWXYZ A 4,-13,0-3,1-1,4-2,0 B -1,12,22,3-1,02,5 C 2,1-1,-10,44,-10,2 D 1,6-3,0-1,41,1-1,4 E 0,01,4-3,1-2,3-1,-1 Player 2 Player 1 Look for the best replies But there are strategies with NO red circles That is, there are Dominated Strategies

Strategic Behavior in Business and Econ Example VWXYZ A 4,-13,0-3,1-1,4-2,0 B -1,12,22,3-1,02,5 C 2,1-1,-10,44,-10,2 D 1,6-3,0-1,41,1-1,4 E 0,01,4-3,1-2,3-1,-1 Player 2 Player 1 Look for the best replies But there are strategies with NO red circles That is, there are Dominated Strategies

Strategic Behavior in Business and Econ Example VWXYZ A 4,-13,0-3,1-1,4-2,0 B -1,12,22,3-1,02,5 C 2,1-1,-10,44,-10,2 D 1,6-3,0-1,41,1-1,4 E 0,01,4-3,1-2,3-1,-1 Player 2 Player 1 Look for the best replies We can eliminate the dominated strategies !

Strategic Behavior in Business and Econ Example VWXYZ A 4,-13,0-3,1-1,4-2,0 B -1,12,22,3-1,02,5 C 2,1-1,-10,44,-10,2 Player 2 Player 1 Look for the best replies We can eliminate the dominated strategies !

Strategic Behavior in Business and Econ Example VWXYZ A 4,-13,0-3,1-1,4-2,0 B -1,12,22,3-1,02,5 C 2,1-1,-10,44,-10,2 Player 2 Player 1 Look for the best replies Now there are strategies with NO red circles for Player 2

Strategic Behavior in Business and Econ Example VWXYZ A 4,-13,0-3,1-1,4-2,0 B -1,12,22,3-1,02,5 C 2,1-1,-10,44,-10,2 Player 2 Player 1 Look for the best replies Now there are strategies with NO red circles for Player 2

Strategic Behavior in Business and Econ Example VWXYZ A 4,-13,0-3,1-1,4-2,0 B -1,12,22,3-1,02,5 C 2,1-1,-10,44,-10,2 Player 2 Player 1 Look for the best replies We can eliminate, again, the Dominated Strategies

Strategic Behavior in Business and Econ Example XYZ A -3,1-1,4-2,0 B 2,3-1,02,5 C 0,44,-10,2 Player 2 Player 1 Look for the best replies We can eliminate, again, the Dominated Strategies

Strategic Behavior in Business and Econ Example XYZ A -3,1-1,4-2,0 B 2,3-1,02,5 C 0,44,-10,2 Player 2 Player 1 Look for the best replies Now we find new Dominated Strategies

Strategic Behavior in Business and Econ Example XYZ A -3,1-1,4-2,0 B 2,3-1,02,5 C 0,44,-10,2 Player 2 Player 1 Look for the best replies Now we find new Dominated Strategies

Strategic Behavior in Business and Econ Example XYZ B 2,3-1,02,5 C 0,44,-10,2 Player 2 Player 1 Look for the best replies The process of elimination of Dominated Strategies continues

Strategic Behavior in Business and Econ Example XYZ B 2,3-1,02,5 C 0,44,-10,2 Player 2 Player 1 Look for the best replies The process of elimination of Dominated Strategies continues

Strategic Behavior in Business and Econ Example XZ B 2,32,5 C 0,40,2 Player 2 Player 1 Look for the best replies The process of elimination of Dominated Strategies continues

Strategic Behavior in Business and Econ Example XZ B 2,32,5 C 0,40,2 Player 2 Player 1 Look for the best replies The process of elimination of Dominated Strategies continues

Strategic Behavior in Business and Econ Example XZ B 2,32,5 Player 2 Player 1 Look for the best replies The process of elimination of Dominated Strategies continues

Strategic Behavior in Business and Econ Example XZ B 2,32,5 Player 2 Player 1 Look for the best replies The process of elimination of Dominated Strategies continues

Strategic Behavior in Business and Econ Example Z B 2,5 Player 2 Player 1 Look for the best replies The process of elimination of Dominated Strategies continues

Strategic Behavior in Business and Econ Example Z B 2,5 Player 2 Player 1 Look for the best replies The final (predicted) outcome of the game is Player 1 chooses B and gets a payoff of 2 Player 2 choose Z and gets a payoff of 5 (Notice that this was the only outcome with “coincidence” of red circles)

Strategic Behavior in Business and Econ The order of elimination of Dominated Strategies does not affect the final outcome of the process Sometimes the process of elimination continues until a unique outcome survives, sometimes it stops earlier If some outcome of the game has a “coincidence of red circles”, then it will survive the process of elimination

Strategic Behavior in Business and Econ Nash Equilibrium after John F. Nash Jr. (1928-) There are (many) games in which the two previous solution concepts can not be used. That is, there (many) games with no Dominant Strategies nor Dominated Strategies. The ideal would be to have a solution concept that can be used in any game and that always works. That is the “Nash Equilibrium”

Strategic Behavior in Business and Econ 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

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 Battle of the Sexes Pat and Chris want to go out together after work They work on different places and before going to work they couldn't find any agreement on where to go The options were go to the Opera of go to the Football They both would like to go to a place together, but Pat prefers the Opera whereas Chris likes the Football better Thus, the situation is that after work (5 pm) each must decide where to go without knowing the choice of the other

Strategic Behavior in Business and Econ The environment of the game Players:Pat and Chris Strategies:Opera or Football Payoffs:In this case we must “define” the payoffs in such a way that represent the game described (see the Table in the next slide) The Rules of the Game Timing of movesSimultaneous Nature of conflict and interactionCoordination Information conditionsSymmetric

Strategic Behavior in Business and Econ The game represented Football Opera Pat Chris 3, 1 Football Opera 0, 0 1, 3 -1, -1 Look for the best replies

Strategic Behavior in Business and Econ The game represented Football Opera Pat Chris 3, 1 Football Opera 0, 0 1, 3 -1, -1 Look for the best replies

Strategic Behavior in Business and Econ The game represented Football Opera Pat Chris 3, 1 Football Opera 0, 0 1, 3 -1, -1 There are two Nash Equlibria

Strategic Behavior in Business and Econ The game represented Football Opera Pat Chris 3, 1 Football Opera 0, 0 1, 3 -1, -1 Pat and Chris going both to the Opera is stable

Strategic Behavior in Business and Econ The game represented Football Opera Pat Chris 3, 1 Football Opera 0, 0 1, 3 -1, -1 Pat and Chris going both to the Football is stable

Strategic Behavior in Business and Econ The game represented Football Opera Pat Chris 3, 1 Football Opera 0, 0 1, 3 -1, -1 Any other outcome is unstable

Strategic Behavior in Business and Econ The game represented Football Opera Pat Chris 3, 1 Football Opera 0, 0 1, 3 -1, -1 This game calls for coordination

Strategic Behavior in Business and Econ Example: The Rock-Paper-Scissors Game RockPaperScissors Player 1 Rock0, 0-1, +1+1, -1 Paper +1, -10, 0-1, +1 Scissors -1, +1+1, -10, 0 Player 2

Strategic Behavior in Business and Econ RockPaperScissors Player 1 Rock0, 0-1, +1+1, -1 Paper +1, -10, 0-1, +1 Scissors -1, +1+1, -10, 0 Player 2 Look for the best replies Example: The Rock-Paper-Scissors Game

Strategic Behavior in Business and Econ RockPaperScissors Player 1 Rock0, 0-1, +1+1, -1 Paper +1, -10, 0-1, +1 Scissors -1, +1+1, -10, 0 Player 2 There are not “always good” nor “always bad” strategies. Example: The Rock-Paper-Scissors Game

Strategic Behavior in Business and Econ RockPaperScissors Player 1 Rock0, 0-1, +1+1, -1 Paper +1, -10, 0-1, +1 Scissors -1, +1+1, -10, 0 Player 2 And there is no coincidence of red circles. !! There is no stable outcome ! Example: The Rock-Paper-Scissors Game

Strategic Behavior in Business and Econ RockPaperScissors Player 1 Rock0, 0-1, +1+1, -1 Paper +1, -10, 0-1, +1 Scissors -1, +1+1, -10, 0 Player 2 In this games, players choose their strategy “at random” This will be the equilibrium Example: The Rock-Paper-Scissors Game

Strategic Behavior in Business and Econ Nash Equilibria are stable outcomes of the game The concept of Nash Equilibrium does not tell how that outcome is reached In case of more than one Nash Equilibria, we do not know which one will be the one chosen by the players (Advanced Game Theory has more solution concepts to “select” among several Nash Equilibria) An Equilibrium in Dominant Strategies is also a Nash Equilibrium A Nash Equilibrium survives the process of elimination of Dominated Strategies A Nash Equilibrium is a one-for-all solution A Nash Equilibrium always exists, in any game

Strategic Behavior in Business and Econ Summary If you have a Dominant Strategy use it, and expect your opponent to use it as well If you have Dominated Strategies do not use any of them, and expect you opponent not to use them as well (eliminate them from the analysis of the game) If there are neither Dominant Strategies nor Dominated Strategies, look for Nash Equilibria and play accordingly. Expect your opponent to play according to the Nash Equilibrium as well If there are multiple Nash Equilibria, further analysis is required