Normal Form Games, Rationality and Iterated Deletion of Dominated Strategies

1. Normal Form Games, Rationality and Iterated Deletion of Dominated Strategies Instructor: Professor Piotr Gmytrasiewicz Presented By: BIN WU Date:11/20/2002

2. Contents • Definition • Typical Normal Form Games • Rational Behavior • Iterated Dominance • Cournot Competition

3. Definition • A Normal Form Game is a game of complete information in which there is a list of n players, numbered 1, 2, … n. Each player has a strategy set, Si, and a utility function • In such a game each player simultaneously selects a move si Si and receives Ui((s1, s2,….)).

4. Componts of normal form game • A list of players D={1,2,….n} • A list of finite strategy sets {S1, S2,…Sn} • Set of strategy profiles S=S1 S2 … Sn • Payoff functions ui: S1S2 ..  Sn R (i =1, 2 .. n)

5. Typical normal form games Normal form games with two players and finite strategy sets can be represented in normal form, a matrix where the rows each stand for an element of S1 and the column for an element of S2. Each cell of the matrix contains an ordered pair which states the payoffs for each player. That is, the cell i, j contains (u1(si, sj), u2(si, sj)) where si is the i-th element of S1 and sj is the j-th element of S2.

6. (1, -1) (-1,1) (-1, 1) (1, -1) Zero_sum games Head Tail Head Tail

7. Zero_sum games • Players: 1, 2 • Strategy sets: {Head, Tail}, {Head, Tail} • Strategy profiles: (Head, Head), (Head, Tail), (Tail, Head), (Tail, Tail).

8. Payoff Functions Payoff functions: • u1(Head, Head) = 1, u1(Head, Tail) = -1, u1(Tail, Head) = -1, u1(Tail, Tail) = 1 • u2(Tail, Head) = 1, u2(Tail, Tail) = -1 u2(Head, Head) = -1, u2(Head, Tail) = 1,

9. (2, 1) (0, 0) (0, 0) (1, 2) Battle of the Sexes Football Opera Football Opera

10. Battle of the Sexes • Where • Husband selections are rows • wife’s are columns

11. (-1, -1) (-10,0) (0, -10) (-3, -3) prisoner's dilemma Cooperate Defect Cooperate Defect

12. prisoner's dilemma • Intuition:I would choose Defect to avoid 10 years of prison • Note: This is the most famous example of Normal Form Games

13. Cournot Competition Two firms each chooses output level qi to maximize his profit, the price of a single product is determined by the total output of the two firms, i.e., p(q1+q2) and each firm suffers the cost ci(qi).

14. Cournot Competition • Players list: D= {1, 2} • Strategy sets: S1 = S2 = R+ • Utility functions: u1(q1, q2) = q1 p(q1, q2)-c1(q1) u2(q1, q2) = q2 p(q1, q2)-c2(q2)

15. Rational Behavior • What is a rational behavior? • The answer depends on my beliefs of my opponent’s actions and my decisions!

16. Rational Behavior • Definition • Player i performs a rational strategy si with beliefs i if where s-i denotes a profile of strategy choices of all other players

17. Rational Behavior For example in the prisoner’s dilemma, suppose I am player 1 and if my beliefs of my opponent’s behaviors are 1 (Cooperate) = 0.5 1 (Defect) = 0.5

18. Rational Behavior • If I choose to cooperate, my expected payoff will be: • u1(Cooperate, Cooperate) 1Cooperate) • + u1(Cooperate, Defect) 1 (Defect) • = -1  0.5 + (-10)  0.5 • = -5.5

19. Rational Behavior • If I choose to defect, then my expected payoff will be: • u1(Defect, Cooperate) 1(Cooperate) • + u1(Defect, Defect) 1 (Defect) • = 0  0.5 + (-3)  0.5 • = -1.5 •  Thus the rational behavior of mine would be to Defect based on my belief functions

20. Dominated Strategies Definition: Strategy si is strictly dominated for player i if there is some si’ Si such that ui(si’, s-i) > ui(si, s-i) For all s-i  S-i . Based on above definition, a rational player i should not choose si no matter what his beliefs are.

21. (2, 2) (1, 1) (4, 0) (1, 2) (4, 1) (3, 5) Iterated Dominance L M R U D

22. Iterated Dominance • If player 1 and player 2 are both rational players and they both know that the other is. • Player 2 should never choose action M because M is dominated. •   Player 1 knows that player 2 is rational

23. (2, 2) (4, 0) (1, 2) (3, 5) Iterated Dominance L R U D

24. (2, 2) (4, 0) Iterated Dominance • Player 1 never chose action D because D is dominated. • Player 2 knows that player 1 is rational L R U

25. Iterated Dominance As a rational player, player 2 chooses L. A “Rational” game yields the result (U, L).

26. Iterated Dominance Deletion Algorithm • Step 1 Define: • Step 2 Define:

27. Iterated Dominance Deletion Algorithm • Step k+1:define: • Step: Let

28. Iterated Dominance Deletion Algorithm The computation must stop after finite number of steps if the strategy sets are finite. An example of Iterated Dominance Deletion:

29. (5, 2) (2, 6) (1, 4) (0, 4) (0, 0) (3, 2) (2, 1) (1, 1) (7, 0) (2, 2) (1, 5) (5, 1) (9, 5) (1, 3) (0, 2) (4, 8) Iterated Dominance Deletion Algorithm A B C D A B C D

30. Iterated Dominance Deletion Algorithm Solution with Iterated Dominance Deletion: • Step1: • S10 = {A, B, C, D} • S20 = {A, B, C, D} • Step 2: • S11 = {A, B, C, D} • S21 = {B, C, D} (A dominated by D)

31. (2, 6) (1, 4) (0, 4) (3, 2) (2, 1) (1, 1) (2, 2) (1, 5) (5, 1) (1, 3) (0, 2) (4, 8) Iterated Dominance Deletion Algorithm B C D A B C D

32. Iterated Dominance Deletion Algorithm • Step3: • S12 = {B, C} (A dominated by B, D dominated by C) • S22 = {B, C} (D dominated by B)

33. (3, 2) (2, 1) (2, 2) (1, 5) Iterated Dominance Deletion Algorithm B C B C

34. Step 4: S13 = {B} (C dominated by B) S23 = {B} (C dominated by B) The resulting strategy profile is (B, B). Luckily, this problem is solvable with IDD. Iterated Dominance Deletion Algorithm

35. Iterated Dominance Deletion Algorithm Definition: G is solvable by pure Iterated Deletion of Strict Dominance if Scontains a single strategy profile.

36. Iterated Dominance Deletion Algorithm Why not weak dominance deletion? If a game is solvable by strict dominance deletion, a consistent strategy profile is generated regardless of the order you eliminate strategies; however, weak dominance deletion may yield different results if you choose different orders. See the following example:

37. (1, 1) (0, 0) (1, 1) (2, 1) (0, 0) (2, 1) Iterated Dominance Deletion Algorithm L R T M B

38. Iterated Dominance Deletion Algorithm • -if we first delete T then L, the final output of utilities will be nothing other than (2, 1) • -  if we first delete B then R, the final utilities will be (1, 1).

39. Cournot Competition Two firms each chooses output level qi to maximize his profit, the price of a single product is determined by the total output of the two firms, i.e., p(q1+q2) and each firm suffers the cost ci(qi).

40. Cournot Competition We can use the Iterated Strict Dominance Deletion to obtain a maximum profit strategy profile for the two competitive firms.

41. Cournot Competition • Assume the market price is determined by the following function: • Assume the cost per product is a constant c for both firms

42. Cournot Competition • The profits for firm 1 and firm 2 are • To achieve the maximum profit, each firm must satisfy the first-order derivative condition:

43. Cournot Competition And

44. Cournot Competition

45. Cournot Competition • we denote q1 and q2 computed above as the “best response function” of the opponent’s output level: q1 = BR(q2) and q2 = BR(q1). • Now we perform the Iterated deletion:

46. Cournot Competition • Step1: both firms can set any output level: • S10 = S20 = R+ • Step2: • S11 = S21 = [0, (-c)/2] This is because each firm knows that his opponent has an output equal to or greater than 0, each firm must select a strategy within this range.

47. Cournot Competition • Step3: • Let’s denote 0 as q- and (-c)/2 as q+, since each firm knows the other’s output is in the range [q-, q+], he must narrow his strategy set to [BR(q+), BR(q-)]—any strategy outside of this range will for sure be strictly dominated by one inside. Thus

48. Cournot Competition • S12 = S22 = [BR(q+), BR(q-)] • Step k: • Iterate until S1k and S2k converge to a same point, q. • The strategy profile (q, q) is the solution generated by the Iterated Deletion of Strict Dominance.

49. A example of Cournot Competition Two companies both produce personal computers, let  = $5000 (a price for the first available PC on the market), =0.5 (free if the total output reaches 10000), c = $895 (the cost is really cheap). Let’s randomly choose s10= 100 and s20=200 (because the next step will guarantee the strategy sets to fall in the range [q-, q+]). The Iterated Deletion of Strict Dominance yields the following result:

50. A example of Cournot Competition [4005.00, 2102.50] [3053.75, 2578.13] [2815.94, 2697.03] [2756.48,2726.76] [2741.62, 2734.19] [2737.91, 2736.05] [2736.98, 2736.51] [2736.74, 2736.63] [2736.69, 2736.66] [2736.67, 2736.66] [2736.67, 2736.67] [2736.67, 2736.67] Well, to produce 2737 PCs each will be the best choice !