On Taxed Matrix Games and Changes in the Expected Transfer

In gambling scenarios the introduction of taxes may affect playing behavior and the transferred monetary volume. Using a game theoretic approach, we ask the following: How does the transferred monetary volume change when the winner has to pay a tax proportional to her win? In this paper we therefore introduce a new parameter: the expected transfer. For a zerosum matrix game with payoff matrix and mixed strategies and of the two players it is defined by . Surprisingly, it turns out that for small fair matrix games higher tax rates lead to an increased expected transfer. This phenomenon occurs also in analogous situations with tax on the loser, bonus for the winner, or bonus for the loser. Higher tax or bonus rates lead to overproportional expected revenues for the tax authority or overproportional expected expenses for the grant authority, respectively. 1. Introduction We analyze how taxes change the character of a fair zerosum matrix game with payoff matrix . For mixed strategies and of the two players Max and Min, the expected value of with respect to is . Sometimes one may not be interested in this value but instead in the expected transfer of . We define this new parameter by . The only difference between value and transfer is that in the payoff case the true are taken, and in the transfer case the absolute values . Regarding the matrix entries as monetary units that are transferred from one player to the other (where the sign determines the direction), the expected transfer expresses how many units are transferred in the average, independent of the direction. This new parameter is an interesting one as it provides information about the player’s behavior. The expected transfer is connected to the average stakes of the players. The higher the expected transfer, the higher the average stakes. Moreover, the expected transfer is an interesting parameter for tax authorities, as the expected tax revenue is directly coupled to it. When is a matrix game with unique optimal strategies and , we shortly write . We investigate how changes when the game is changed in a certain way. Our basic model assumes a “winner tax,” say with rate. When row and column are played, then is transferred. For the winner tax means that player Max receives only , whereas player Min has to pay the full . For player Max has to pay the full , but Min gets only . For nothing at all is transferred and no taxes are taken from the players. We denote the new game by , where the index “WiT” is short for “Winner Tax.” Introducing such a tax changes the character of the game. It is no longer