Creating a Payoff Matrix

Date: 06/04/2003 at 20:44:04
From: Elena
Subject: Game theory problem: how to create a payoff matrix

Shoe Town and Fancy Foot are both vying for more share of the market. 

If Shoe Town does no advertising, it will not lose any share of the 
market as long as Fancy Foot does nothing. Shoe Town will lose 2% of 
the market if Fancy Foot invests $10,000 in advertising, and it will 
lose 5% of the market if Fancy Foot invests $20,000 in advertising. 

On the other hand, if Shoe Town invests $15,000 in advertising, it 
will gain 3% of the market as long as Fancy Foot does nothing; it will 
gain 1% of the market if Fancy Foot invests $10,000 in advertising, 
and it will lose 1% if Fancy Foot invests $20,000 in advertising.

(A) Develop a payoff table for this problem.
(B) Determine the various strategies.

                Fancy Foot
             Y1     Y2     Y3
Shoe  X1    -2%    -5%      0
Town  X2     1%    -1%      3

Date: 06/04/2003 at 21:30:32
From: Doctor Shawn
Subject: Re: Game theory problem: how to create a payoff matrix

Elena,

Your payoff matrix is a good attempt, but keep in mind that each
player in this 2-player game has a payoff from each endpoint of the
game. 

Shoe Town has two pure strategies: do nothing (X1), or invest $15,000 
in advertising (X2). Fancy Foot has three strategies: do nothing (Y1), 
invest $10,000 in advertising (Y2), or invest $20,000 in advertising 
(Y3). There are only two shoe stores in town, so if one of them gains 
market share, the other one has to lose it. This is called a zero-sum 
game.

The revised payout table looks like this, with payoffs of (Shoe Town,
Fancy Foot):

                   Fancy Foot
           Y1           Y2            Y3
S. X1   (+0,-0)      (-2%,+2%)    (-5%,+5%)
T. X2   (+3%,-3%)    (+1%,-1%)    (-1%,+1%)

Now we want to find if there is a Nash equilibrium in pure strategies.

Look first to Shoe Town. If they play strategy X2, they always do 
better than if they play X1, no matter what Fancy Foot does. That 
means that X1 is a _strongly dominated_ strategy, and no rational
player will ever choose to play X1. Fancy Foot knows this, so they'll
choose the strategy that will give them the most payoff in X2, namely
Y3. Therefore, the Nash equilibrium for this game is (X2, Y3) and both 
players spend their maximum advertising revenue.

This brings up a very interesting and subtle point. The payoffs in a
game like this are usually measured in "utils" and not money or market
share. The point of that is that you want to use something of equal
value to both players, even if it's not necessarily the same amount of
stuff. For instance, $100 is not worth as much to Ted Turner as it is
to me. As a result, you should be careful about making conclusions in
this game. I was assuming that the payoff in market share was worth
any cost to the players, but in reality losing 2% of market share
might actually be worth less than $10,000. If that's the case, then
the players will choose different strategies.

I hope that helps!

- Doctor Shawn, The Math Forum
  http://mathforum.org/dr.math/