Analyzing a game for fairness? (game theory question)

[musicgold]

Hi,

Please see the attached picture. Table 1 in the picture shows the payoffs of a 2-player game, similar to the Rock Paper Scissor game. In this game, the two player (Stephen and Maude) show either one or two fingers at the count of three. If the sum of the total fingers shown is an even number Stephen wins and Maude pays him a number of dollars equal to the sum. For example if both of them show 1 finger each, Stephen gets $2 from Maude. If the sum is an odd number, Stephen has to pay the amount to Maude. The first number in each cell is Stephen’s payoff and the second number is Maude’s payoff.

I am trying to find a way to figure if the game is fair or biased towards one player. How can I do that just by looking at such a payoff table.

Table 2 is the payoff table for the game in which the two players can show up to three fingers. How should I analyze a game like that for fairness?


Thanks.

 

[kai_sikorski]

For the first game I would just assume that one player choses with uniform probability among his options. Then parametrize the second players possible strategies by the one number p, to choose the 1 finger (choosing 2 fingers is then done with probability 1-p). You should be able to analyze the payoff in this way for an arbitrary strategy for player-2. And I think you'll find that the expected payoff is always 0. So basically player-1's strategy can not be exploited. I haven't done the calculation but I think that's what you'll find. So this game is fair.

So basically if you do think the game is fair, and you think you know what the un-exploitable strategy is then the problem isn't that hard because you can just assume this strategy for one of the players and cary through the consequences.

If you think that the game is NOT fair, then you have to show that for ever possible strategy for one of the players, the other can exploit it. So that's a bit harder.

 

[SW VandeCarr]

An example of a fair game is a martingale. In such a game, prior events have no bearing on current events. One player cannot obtain an advantage by any strategy where knowing the outcome of a prior event increases his/her probability of winning at later point in time.

Formally, for a sequence of random variables $x_1, x_2,...,x_n$ the conditional expectation for each $x_{n+1}$ is $x_n$

 

[musicgold]

I know for sure that the game is not fair. It is biased towards Maude.

 

[kai_sikorski]

Let p and q be the probability for Maude and Stephen to put up 1 finger respectively.

Maude's expected payoff is given by

E[M] = q (p *2 + (1 - p)*(-3)) + (1 - q) (p*(-3) + (1 - p)*4) = 4 - 7 q + p (-7 + 12 q)

Suppose Stephen puts 1 finger up with probability 7/12

E[M] = - 1/12

So you're right it is biased, but actually not toward Maude. You can do the second scenario the same way. Parametrize the strategies, and look for parameters for Stephen (or Maude) such that Maude's (or Stephens) expectation does not depend on Maude's (or Stephen's) strategy.

EDIT: ups I got the indecies for Maude/Stephens payoffs backwards. Yes you're right it's biased towards Maude.

 

[kai_sikorski]

I did the calculation for the second game, because I was curious. If stephen goes .25, .5 and .25 in 1,2 and 3 fingers respectively, then no matter what Maude does the expectation is 0. You can see this easily by doing a weighted average of the rows.

 

[musicgold]

Thanks folks.
It seems there is an even simpler way to figure that out.

You can do the second scenario the same way. Parametrize the strategies, and look for parameters for Stephen (or Maude) such that Maude's (or Stephens) expectation does not depend on Maude's (or Stephen's) strategy.

I am not sure I understand what you mean by 'parametrize'. Can you please explain a bit?

Thanks.