Abstract
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 A and mixed strategies p and q of the two players it is defined by ET(A;p,q)=∑∑piqj|aij|. 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.
Highlights
We analyze how taxes change the character of a fair zerosum matrix game Γ with payoff matrix A =
We investigate how ET(Γ) changes when the game Γ is changed in a certain way
What change in player behavior do we see in real-world games, for example, in the Internet Poker servers with different types and amounts at stake? Or what about actors participating in online betting portals where profit taxes are collected?
Summary
At least in the case, where the taxed game has a unique equilibrium, one may compare the expected transfers of ΓWiT(x) and Γ We do this comparison with respect to the original |aij|. When, for instance, in the taxed game, player Max gets (1 − x) ⋅ a and Min has to pay 1 ⋅ a, we count this as transfer of size 1 ⋅ a. Fair 2 × 2 matrix games and symmetric 3 × 3 matrix games, for every nondegenerate matrix game Γ, the tax leads to an increased expected transfer. For fair 2 × 2 and symmetric 3 × 3 matrix games, all four scenarios lead to larger expected transfers for increasing rates x.
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