Games with Costly Winnings

We introduce a new two-person multi-stage game in which players compete to win a maximal number of stages. At the start of the game each player is given resources which are needed for winning the competitions. After each stage the winner pays an additional maintenance fee for his winning and the size of the fee may depend on the stage. Thus the limited resources of players are needed both for increasing the probability of winning and for paying the fees. We show that in this generality the game may have highly non-unique Nash equilibria. We introduce certain restrictions on the game which guarantee a unique Nash equilibrium, and that the resources will not run out before the end of the game. We then generalize the game and results to m players.