finding saddle point of 2x3 zero sum game

[leo.k...@gmail.com]

Hello

How to find it when there is no solution in pure strategies? For example this game:

1 -1
-1 0
0 1

first player's strategy: [x, y, 1-x-y],
second player's strategy: [u, 1-u]

Of course, x,y,u are nonnegative, strictly smaller than 1.

Then the payoff is: 3ux-2x-y-1. We search for a saddle point (x*,u*).

Variable u needs to be in interior, it can't be 0 or 1, because there are no pure solutions. So I differentiate payoff with regard to u and conclude 3x=0. This can happen.

But then the payoff is: 1-y. It looks like whatever player 2 chooses it's the same and y should be 0, which is impossible. What's wrong?


Thanks in Advance
Lev

[quasi]

leo.krupski wrote:
>
>Hello
>
>How to find it when there is no solution in pure strategies?
>For example this game:
>
>1 -1
>-1 0
>0 1
>
>first player's strategy: [x, y, 1-x-y],
>second player's strategy: [u, 1-u]
>
>Of course, x,y,u are nonnegative, strictly smaller than 1.

Note that row 2 can be eliminated since it's dominated by
row 3. Had you taken advantage of that, the game would have
been reduced to a 2x2 game, still zero sum.

>Then the payoff is: 3ux-2x-y-1.

No, your error is right there -- check your algebra.

quasi

[leo.k...@gmail.com]

Hello and thank you for your answer. But still I have two questions:

1. Then I get x=1/3 for example. OK, so x isn't equal to zero, it is in interior. So then I say: differential of payoff with respect to x is zero. Right?

2. When there are 3 vectors of size 1x2 is this always that one of them is smaller or equal than some convex combination of two remaining? Or if this vectors are all nonnegative?

Best regards
Lev

[quasi]

leo.krupski wrote:

>Hello and thank you for your answer. But still I have two
>questions:
>
>1. Then I get x=1/3 for example. OK, so x isn't equal to zero,
>it is in interior. So then I say: differential of payoff with
>respect to x is zero. Right?

Yes.

>2. When there are 3 vectors of size 1x2 is this always that one of them is smaller or equal than some convex combination of two remaining?

Yes.

>Or if this vectors are all nonnegative?

Irrelevant.

quasi

[leo.k...@gmail.com]

OK. Thank you very much then.

[quasi]

quasi wrote:
>
>leo.krupski wrote:
>
>>Hello and thank you for your answer. But still I have two
>>questions:
>>
>>1. Then I get x=1/3 for example. OK, so x isn't equal to zero,
>>it is in interior. So then I say: differential of payoff with
>>respect to x is zero. Right?
>
>Yes.

I take that back.

It's true in a 2x2 game with no optimal pure strategies.

Hence, since this game can be reduced (after eliminating
row 2) to a 2x2 with no optimal pure strategies, it's true
in this game.

[quasi]

quasi wrote:
>
>quasi wrote:
>>
>>leo.krupski wrote:
>>
>>>Hello and thank you for your answer. But still I have two
>>>questions:
>>>
>>>1. Then I get x=1/3 for example. OK, so x isn't equal to zero,
>>>it is in interior. So then I say: differential of payoff with
>>>respect to x is zero. Right?
>>
>>Yes.
>
>I take that back.
>
>It's true in a 2x2 game with no optimal pure strategies.
>
>Hence, since this game can be reduced (after eliminating
>row 2) to a 2x2 with no optimal pure strategies, it's true
>in this game.

I'll try to clarify ...

In order for the derivative tests to yield necessary conditions
for optimal strategies, I think an optimal strategy has to be
an interior point for _both_ players -- that is, any optimal
strategy must be such that _all_ pure strategies (for both
players) have positive probabilities.

But that can only happen for a _square_ matrix game.

Note that the game in question is 3x2, but can be reduced
to 2x2 after eliminating row 2.

But I'm not so sure about my answer.

Maybe someone more knowledgeable about this can put it right.

quasi