# Zero-Sum Game Theory

Me and a team of programmers are programming a robot to play a game in a competition. We did not create the hardware, and all teams will use the same type of robot.

GAME DESCRIPTION

The game has two players opposing each other and are both trying to achieve an objective. This objective is to move to a position to pick up a payload then return to another position to return the payload. Each team has their own payload and their own target to bring them to, and it is impossible to pick up the other team's payload. The team that brings their payload to the target first wins and the game is ended. There is a time limit of 210 seconds, and if the games times out, the team who held the payload the longest wins.

However, it is a little more complicated than that. The robots also have the ability to face the opponent and "push them back" from any range (the closer they are the more forcefully they push). Also, if a robot is pushed out of the bounds of the playing field, they drop their payload and it is moved back inside the playing field.

GAME THEORY QUESTIONS

First of all, is this a zero-sum game or not? I am new to game theory math and I am not completely sure if it is.

Also, how does the minimax theorem apply to this specific game? I understand how it works, but I do not know what the values to account for would be in this game (would it be the difference in how long each team has held the payload somehow combined with how close they are to bringing to the target?)

I really am not sure at all how to calculate any of this, but if anyone can at least point me in the right direction for coming up with an effective strategy system for this I would be very appreciative.

Thank you so much for your time, this project is very important to me and your help means a lot. If I need to clarify anything please ask, I am not sure if I included all the information needed for this problem.

-

Yes, this is a zero sum game, since both players are fighting over a single "win". What that means in terms of strategy is that hurting your opponent is as effective as helping yourself.

I don't see how you would use minmax theory in this situation. As far as I know, game theory tools are geared towards games with a relatively small number of possible situations and moves, whereas in your game, you have a very large number of possible situations (because the position of the robots is variable).

-

This is a zero sum game differential game.

Your strategy is a function from the state of the board (everyone's position and velocity and how long they have held their payload) to your control inputs. Your payoff is 1 if you win and 0 if you lose. [Which is why this is a zero sum game, since the sum of payoffs over all players equals 1].

Having said that, the actual solved examples of differential games I have seen have been for simple two player problems. You may be better off using some kind of heuristic for the contest.

-