Finding Equilibrium in a Competitive Game with Multiple Choice Variables

[ruzbayhhi]

Homework Statement

I have been grappling with this one for a while now. I have two players in a competitive game. Player A chooses how much to invest in x and y (he has two choice variables), and player B chooses how much to invest in z. Their payoff functions are functions of their choices - (U(A) = f(x,y,z); U(B)=f(x,y,z)). How do I find the equilibrium?

Homework Equations

U(A) = f(x,y,z);
U(B) = f(x,y,z);

The Attempt at a Solution

I have no idea what procedure to follow here..

 

[Mark44]

Homework Statement

I have been grappling with this one for a while now. I have two players in a competitive game. Player A chooses how much to invest in x and y (he has two choice variables), and player B chooses how much to invest in z. Their payoff functions are functions of their choices - (U(A) = f(x,y,z); U(B)=f(x,y,z)). How do I find the equilibrium?

Homework Equations

U(A) = f(x,y,z);
U(B) = f(x,y,z);

The Attempt at a Solution

I have no idea what procedure to follow here..

What does U(A) represent? The payoff to player A?
What does U(B) represent? The payoff to player B?

Is the payoff to each player a function of all three variables?

 

[ruzbayhhi]

Yes, exactly - the payoff function for player A is u(A) and for player B it's u(B).


Player A has two choices to make and Player B has one. To illustrate: the players are at a fighting over a sum of money. Player A can give player B all or some of his money, or invest some or all of his money in fighting. Player B chooses how much to invest in fighting. Player A wants to maximize his payoff (paying some of the money to B would decrease B's incentive to fight over the rest) and player B wants to maximize his.

 

[ruzbayhhi]

Anyone? (It's a bit urgent and I really need your help! CHEERS!)

 

[Mark44]

Yes, exactly - the payoff function for player A is u(A) and for player B it's u(B).


Player A has two choices to make and Player B has one. To illustrate: the players are at a fighting over a sum of money. Player A can give player B all or some of his money, or invest some or all of his money in fighting. Player B chooses how much to invest in fighting. Player A wants to maximize his payoff (paying some of the money to B would decrease B's incentive to fight over the rest) and player B wants to maximize his.

I'm still not crystal clear what's involved here. At the beginning, does player A have all the money and player B wants some of it? Obviously, though, player B must have some money or resources, since it will cost him in some way to fight A for the money that A has.

It also seems to me that player B has a couple of choices, as well. He can choose to fight A for the money, or he can decide that it would cost him more in resources than he would gain by fighting A.

 

[ruzbayhhi]

To clarify I will give a numerical example. At time 1, player A has a $1000 Dollars. Player B wants $800. Player A can choose to pay it all to B, pay none and fight with B (and then he has to decide how much to invest in fighting) and pay some of it and fight over the rest. Player B has one choice - how much to invest in fighting (which increases his probability of winning). As you pointed out he can also decide to invest 0 in fighting, which effectively means he will get nothing.
I do not know how to proceed - how can I solve and find the equilibrium in this case, which involves two choices for one player and one choice for the second?

 

[Mark44]

This seems like a silly game to me. You have one player with money, and another player intent on taking it from the first player. The second player is essentially mugging the first player.

There don't seem to be any consequences for the second player -- as you have described things, the second player either wins something or nothing at all. Unlike the first player, he doesn't lose anything. To make this more interesting, the second player should have the prospect of losing something.

I think that you need to have (or define) probabilities of winning for each player, based on the amount of money being contested. Using your example, if player A has $1000 and opts to give none of it to player B, and player B wants $800, they need to fight or flip a coin or do something to determine the outcome. The probability of one player winning should be affected by how much that player

 

[ruzbayhhi]

I probably should have stressed that fighting over the money is probabilistic process, that is affected by the relative investment of both parties (the more A invests, relative to B, the higher his chances of winning). Given that, Player B stands to lose his investment in fighting and to gain the contested amount (800). On the other hand, Player A stands to lose the contested amount or gain it, if he wins the fight. If you would like, I'd be happy to PM you the full specific game, but I thought it may not be necessary as my question is more geared towards the procedure of solving a game like this than the specific game.

 

[Mark44]

I still don't see what B stands to lose. What money/resources does he have? The way I interpreted what you said, player A has some money, and player B wants it, and B has nothing to lose.

 

[ruzbayhhi]

Player B invests a sum of money in fighting. I don't think it is material how much money he starts off with, but we can assume he starts with $10,000. Player B can choose how much of that to invest in fighting. Obviously he will never invest more than the amount at stake (800). However, he might invest less or nothing at all.

If he wins, and Player A didn't pay anything upfront, he gets 800 minus his investment; if he loses, he loses his entire investment.

 

[Mark44]

Player B invests a sum of money in fighting. I don't think it is material how much money he starts off with, but we can assume he starts with $10,000. Player B can choose how much of that to invest in fighting. Obviously he will never invest more than the amount at stake (800). However, he might invest less or nothing at all.

Which means that B can make either of two choices: to fight or not to fight (invest 0).

If he wins, and Player A didn't pay anything upfront, he gets 800 minus his investment; if he loses, he loses his entire investment.

 

[ruzbayhhi]

Yes, right (but the investment choice is not binary - to invest or not to invest, but rather continuous - how much to invest with zero and everything at the limit).

 

[Mark44]

Understood.

Going back to your question in post #1, see if you can come up with a formula for the payoff to A for the two cases (fight/no fight) and the payoff to B for the two cases (fight/no fight).