3.2.1. Static Games of complete information: Dominant

1. Strategic Behavior in Business and Econ 3.2.1. Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies

2. 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 3.2.2. Dynamic Games of complete information: Backward Induction and Subgame perfection

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

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

5. 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

6. 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)

7. 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

8. Strategic Behavior in Business and Econ Example: Game with “always winning” strategy (Dominant Strategy) Prediction of Game Theory: Both have a clear best strategy Advertiseno matter what Philip Morris (player 2) Not advertise Advertise Not advertise 50, 50 20, 60 Reynolds (player 1) 60, 20 30, 30 Advertise

9. Strategic Behavior in Business and Econ $2 $4 $5 Bar 1 $2 10 , 10 14 , 12 14 , 15 $4 12 , 14 20, 20 28 , 15 $5 15 , 14 15 , 28 25, 25 Example: Game with “worse than” strategies (Dominated Strategies) Prediction of Game Theory: There is no “always winning” strategy Bar 2 (in thousands of dollars)

10. Strategic Behavior in Business and Econ $2 $4 $5 Bar 1 $2 10 , 10 14 , 12 14 , 15 $4 12 , 14 20, 20 28 , 15 $5 15 , 14 15 , 28 25, 25 Prediction of Game Theory: But there is a clearly bad strategy: $2 is always worse than $4 Bar 2 (in thousands of dollars)

11. Strategic Behavior in Business and Econ $2 $4 $5 Bar 1 $2 10 , 10 14 , 12 14 , 15 $4 12 , 14 20, 20 28 , 15 $5 15 , 14 15 , 28 25, 25 Prediction of Game Theory: If $2 is removed from the game (it will never be used) then the game is more clear Bar 2 (in thousands of dollars)

12. Strategic Behavior in Business and Econ Prediction of Game Theory: Now $4 is clearly the best strategy no matter what my competitor does Bar 2 $4 $5 Bar 1 $4 20, 20 28 , 15 $5 15 , 28 25, 25 (in thousands of dollars)

13. Strategic Behavior in Business and Econ Prediction of Game Theory: NOTICE: The “coincidence” of red circles is (again) the stable outcome Bar 2 $4 $5 Bar 1 $4 20, 20 28 , 15 $5 15 , 28 25, 25 (in thousands of dollars)

14. Strategic Behavior in Business and Econ Right Left Paul Right 0 , 0 -50 , -50 Left -50 , -50 0 , 0 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 Mary 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!)

15. Strategic Behavior in Business and Econ 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 Player 2 Rock Paper Scissors Rock 0, 0 -1, +1 +1, -1 Player 1 Paper +1, -1 0, 0 -1, +1 Scissors -1, +1 +1, -1 0, 0

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

17. 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

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

19. 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 NOred 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

20. Strategic Behavior in Business and Econ Example Player 2 Player 1

21. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1

22. Strategic Behavior in Business and Econ Example Notice that (B,Z) is the only outcome with coincidence of red circles Look for the best replies Player 2 Player 1 There are no strategies with “all red circles” That is, there are no Dominant Strategies

23. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 But there are strategies with NO red circles That is, there are Dominated Strategies

24. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 But there are strategies with NO red circles That is, there are Dominated Strategies

25. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 We can eliminate the dominated strategies !

26. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 We can eliminate the dominated strategies !

27. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 Now there are strategies with NO red circles for Player 2

28. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 Now there are strategies with NO red circles for Player 2

29. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 We can eliminate, again, the Dominated Strategies

30. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 We can eliminate, again, the Dominated Strategies

31. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 Now we find new Dominated Strategies

32. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 Now we find new Dominated Strategies

33. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 The process of elimination of Dominated Strategies continues

34. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 The process of elimination of Dominated Strategies continues

35. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 The process of elimination of Dominated Strategies continues

36. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 The process of elimination of Dominated Strategies continues

37. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 The process of elimination of Dominated Strategies continues

38. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 The process of elimination of Dominated Strategies continues

39. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 The process of elimination of Dominated Strategies continues

40. Strategic Behavior in Business and Econ Example Look for the best replies Player 2 Player 1 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)

41. 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

42. 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”

43. 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

44. Strategic Behavior in Business and Econ

45. 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)

46. 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

47. 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 moves Simultaneous • Nature of conflict and interaction Coordination • Information conditions Symmetric

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

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

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