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

 Definition  Typical Normal Form Games  Rational Behavior  Iterated Dominance  Cournot Competition

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, S i, and a utility function In such a game each player simultaneously selects a move s i  S i and receives U i ((s 1, s 2,….)).

 A list of players D={1,2,….n}  A list of finite strategy sets {S 1, S 2,…S n }  Set of strategy profiles S=S 1  S 2  …  S n  Payoff functions u i : S 1  S 2 ..  S n  R (i =1, 2.. n)

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 S 1 and the column for an element of S 2. Each cell of the matrix contains an ordered pair which states the payoffs for each player. That is, the cell i, j contains (u 1 (s i, s j ), u 2 (s i, s j )) where s i is the i-th element of S 1 and s j is the j-th element of S 2.

(1, -1) (-1,1) (-1, 1) (1, -1) Head Tail Head Tail

Players: 1, 2 Strategy sets: {Head, Tail}, {Head, Tail} Strategy profiles: (Head, Head), (Head, Tail), (Tail, Head), (Tail, Tail).

Payoff functions: o u 1 (Head, Head) = 1, u 1 (Head, Tail) = -1, u 1 (Tail, Head) = -1, u 1 (Tail, Tail) = 1 o u 2 (Tail, Head) = 1, u 2 (Tail, Tail) = -1 u 2 (Head, Head) = -1, u 2 (Head, Tail) = 1,

(2, 1) (0, 0) (0, 0) (1, 2) FootballOpera Football Opera

Where  Husband selections are rows  wife’s are columns

(-1, -1) (-10,0) (0, -10) (-3, -3) CooperateDefect Cooperate Defect

Intuition: I would choose Defect to avoid 10 years of prison Note: This is the most famous example of Normal Form Games

Two firms each chooses output level q i to maximize his profit, the price of a single product is determined by the total output of the two firms, i.e., p(q 1 +q 2 ) and each firm suffers the cost c i (q i ).

Players list: D= {1, 2} Strategy sets: S 1 = S 2 = R + Utility functions: u 1 (q 1, q 2 ) = q 1  p(q 1, q 2 )-c 1 (q 1 ) u 2 (q 1, q 2 ) = q 2  p(q 1, q 2 )-c 2 (q 2 )

 What is a rational behavior? The answer depends on my beliefs of my opponent’s actions and my decisions!

 Definition Player i performs a rational strategy s i with beliefs  i if where s -i denotes a profile of strategy choices of all other players

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

 If I choose to cooperate, my expected payoff will be: u 1 (Cooperate, Cooperate)  1 Cooperate) + u 1 (Cooperate, Defect)  1 (Defect) = -1  (-10)  0.5 = -5.5

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

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

(2, 2) (1, 1) (4, 0) (1, 2) (4, 1) (3, 5) LMR D U

 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

(2, 2) (4, 0) (1, 2) (3, 5) LR D U

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

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

 Step 1 Define:  Step 2 Define:

 Step k+1:define:  Step  : Let

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

(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) ABC D A B C D

Solution with Iterated Dominance Deletion:  Step1: S 1 0 = {A, B, C, D} S 2 0 = {A, B, C, D}  Step 2: S 1 1 = {A, B, C, D} S 2 1 = {B, C, D} (A dominated by D)

(2, 6) (1, 4) (0, 4) (3, 2) (2, 1) (1, 1) (2, 2) (1, 5) (5, 1) (1, 3) (0, 2) (4, 8) BDC B C D A

 Step3: S 1 2 = {B, C} (A dominated by B, D dominated by C) S 2 2 = {B, C} (D dominated by B)

(3, 2) (2, 1) (2, 2) (1, 5) B C B C

 Step 4: S 1 3 = {B} (C dominated by B) S 2 3 = {B} (C dominated by B) The resulting strategy profile is (B, B). Luckily, this problem is solvable with IDD.

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

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:

(1, 1) (0, 0) (1, 1) (2, 1) (0, 0) (2, 1) LR T M B

 - 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).

Two firms each chooses output level q i to maximize his profit, the price of a single product is determined by the total output of the two firms, i.e., p(q 1 +q 2 ) and each firm suffers the cost c i (q i ).

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

 Assume the market price is determined by the following function:  Assume the cost per product is a constant c for both firms

 The profits for firm 1 and firm 2 are  To achieve the maximum profit, each firm must satisfy the first-order derivative condition:

And

 we denote q 1 and q 2 computed above as the “best response function” of the opponent’s output level: q 1 = BR(q 2 ) and q 2 = BR(q 1 ). Now we perform the Iterated deletion:

 Step1: both firms can set any output level: S 1 0 = S 2 0 = R +  Step2: S 1 1 = S 2 1 = [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.

 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

S 1 2 = S 2 2 = [BR(q+), BR(q-)]  Step k: Iterate until S 1 k and S 2 k converge to a same point, q. The strategy profile (q, q) is the solution generated by the Iterated Deletion of Strict Dominance.

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 s 1 0 = 100 and s 2 0 =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:

[ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] Well, to produce 2737 PCs each will be the best choice !