Von Neumann and Newman poker with a flip of hand values

The von Neumann and Newman poker models are simplified two-person poker models in which hands are modeled by real values drawn uniformly at random from the unit interval. We analyze a simple extension of both models that introduces an element of uncertainty about the final strength of each player’s own hand, as is present in real poker games. Whenever a showdown occurs, an unfair coin with fixed bias q is tossed, 0 ≤ q ≤ 1/2. With probability 1− q, the higher hand value wins as usual, but with the remaining probability q, the lower hand wins. Both models favour the first player for q = 0 and are fair for q = 1/2. Our somewhat surprising result is that the first player’s expected payoff increases with q as long as q is not too large. That is, the first player can exploit the additional uncertainty introduced by the coin toss and extract even more value from his opponent.