Mixed Strategies: Len Guienie and Al Dente Battle for the Spaghetti Market |
Len and Al are locked in a strategic battle for the pasta market in Philadelphia. The game that confronts them is presented in normal form below.
Al Dente | ||||
Left | Center | Right | ||
Len Guienie | Up | 10, 0 | 0, 10 | 3, 3 |
Middle | 2, 10 | 10, 2 | 6, 4 | |
Down | 3, 3 | 4, 6 | 6, 6 |
1. What is the payoff to Al when he plays Left and Len plays Up? That is find u_{Al}(Up_{Len}, Left_{Al}) = 0.
2. What is the payoff to Len for u_{Len}(Middle_{Len}, Right_{Al})? 6
Let s_{i }(*, *, *) denote the probabilities that the i^{th} player plays her available strategies. For example, s_{Len}(1/6, 4/6, 1/6) means that Len plays U with probability 1/6, he plays M with probability 4/6, and plays D with probability 1/6.
3. What is the payoff for Al for u_{Al}(s_{Len}, C) when s_{Len} = (1/3, 2/3 , 0)? u_{Al}(s_{Len},C) = 10(1/3)+2(2/3)+6(0) = 14/3.
4. What is the payoff for Len for u_{Len}((1/3, 2/3, 0),C)? u_{Len}((1/3, 2/3, 0),C) = 0(1/3)+10(2/3)+4(0) = 20/3.
5. Find the payoff u_{Al}(s_{Len},s_{Al}), where s_{Len} = (1/2, 1/2, 0) and s_{Al} = (1/4, 1/4, 1/2). Let's first figure out the joint probability distribution for possible moves made by Al and Len.
Al Dente | |||||
Left | Center | Right | |||
Len Guienie | Up | 1/8 | 1/8 | 1/4 | 1/2 |
Middle | 1/8 | 1/8 | 1/4 | 1/2 | |
Down | 0 | 0 | 0 | 0 | |
1/4 | 1/4 | 1/2 | 1.0 |
Now we can compute the expected payoff for Al.
u_{Al}(s_{1},s_{2}) = 1/8(0)+1/8(10)+0(3) + 1/8(10)+1/8(2)+0(6) + 1/4(3)+1/4(4)+0(6) = 36/8.
6. What mixed strategy should Len play? Suppose Len plays Up with probability p_{1} and M with probability p_{2}. He has to choose these probabilities so that Al is indifferent between any of his own pure strategies.
p_{1}(0)+p_{2}(10)+(1-p_{1}-p_{2})(3) = p_{1}(10)+p_{2}(2)+(1-p_{1}-p_{2})(6)
p_{1}(0)+p_{2}(10)+(1-p_{1}-p_{2})(3) = p_{1}(3)+p_{2}(4)+(1-p_{1}-p_{2})(6)
If you solve the two equations for the two unknowns then p_{1}=2/21 (.095) and p_{2}=1/3 (.333).
7. Let's see if we can find out where the argument that Len chooses p_{1} and p_{2} so that Al is indifferent between his pure strategies 'comes from.' First, we'll construct the joint distribution over outcomes when they are both playing mixed strategies. Len is playing S_{Len}(p_{1}, p_{2}, 1-p_{1}-p_{2}) and Al is playing S_{Al}(q_{1}, q_{2}, 1-q_{1}-q_{2})
Al Dente | |||||
Left | Center | Right | |||
Len Guienie | Up | p_{1}q_{1} | p_{1}q_{2} | p_{1}(1-q_{1}-q_{2}) | p_{1} |
Middle | p_{2}q_{1} | p_{2}q_{2} | p_{2}(1-q_{1}-q_{2}) | p_{2} | |
Down | (1-p_{1}-p_{2})q_{1} | (1-p_{1}-p_{2})q_{2} | (1-p_{1}-p_{2})(1-q_{1}-q_{2}) | 1-p_{1}-p_{2} | |
q_{1} | q_{2} | 1-q_{1}-q_{2} | 1.0 |
Second, compute Len's expected payoff when they are both playing mixed strategies.
10p_{1}q_{1}+2p_{2}q_{1}+3(1-p_{1}-p_{2})q_{1} + 0p_{1}q_{2}+10p_{2}q_{2}+4(1-p_{1}-p_{2})q_{2}+3p_{1}(1-q_{1}-q_{2})+6p_{2}(1-q_{1}-q_{2}) + 6(1-p_{1}-p_{2})(1-q_{1}-q_{2})
Len's task is to choose p_{1} and p_{2} as to maximize his expected payoff. So, the third step is to take the partial derivatives of his expected payoff with respect to p_{1} and p_{2} and set the results equal to zero.
10q_{1}-q_{2}-3 = 0
-q_{1}+6q_{2} = 0
Note that although we set the problem up as though Len were choosing his p's, in the first order conditions we end up solving for Al's q's in a set of equations in which the payoffs belong to Al!
So q_{2} = 3/59 = .051 and q_{1} = 6(3/59) = .305 are the probabilities that ought to be played by Al when Len is choosing his own mixed strategy.
Just to be sure we understand what happened let's do it again, but for Al this time. First, write down Al's expected payoff when they are both playing a mixed strategy.
0p_{1}q_{1}+10p_{2}q_{1}+3(1-p_{1}-p_{2})q_{1} + 10p_{1}q_{2}+2p_{2}q_{2}+6(1-p_{1}-p_{2})q_{2}+3p_{1}(1-q_{1}-q_{2})+4p_{2}(1-q_{1}-q_{2}) + 6(1-p_{1}-p_{2})(1-q_{1}-q_{2})
Second, find the partial derivatives with respect to q_{1} and q_{2}, and set the results equal to zero.
9p_{2}-3 = 0
7p_{1}-2p_{2} = 0
Solving the two equations we have p_{1} = 2/21 and p_{2} = 1/3.
8. These mixed strategies result in a Nash equilibrium since they represent a best response to the mixed strategy being used by the opponent.