Game theory - rent seeking?

Question

I want to prove the following, but first a little about me. I have taken courses in proofs, but I am only familiar with proofs at the introductory level. I am only familiar with the basics of game theory. I understand rent seeking fairly well.

I want to mathematically prove the following behavior: Where two players will compete for some given rent, where if both players bid low (nothing) each will have a 50% chance of attracting this rent (indiferent). If one bids high and the other low, the high bidder receives the rent (1). To attract this rent they need to spend a part of this rent and so the rent is erroded marginally.

Let's say after player one bids high for a net payoff of .9. Player two will bid for a net payoff of .8 and so on all the way down to where the net pay off is 0. In theory, there are an infinite amount of choices between 1 and 0.

I want to formally prove this fact to incorporate into a paper I am writing. Can anyone help me or point me to a good resource I can use

Answer 1

A : 1 : 0
B : 0 : 1
-------------------------peak=1--------------------------------------
A :0.9 : 0.1
B: 0.2 : 0.8
----------------------A+B=1--------------------