Proportional three-person red-and-black games

Consider a three-person game that occurs in stages. The state of the game is given by the integral amounts of chips that the players have, say x=(x1, x2, x3) with M=x1+x2+x3 fixed. At a stage of the game, player i places ai chips in the pot, an integer between 1 and xi. (Player i is already eliminated from the game if xi=0.) The winner of the pot is then immediately chosen in such a way that player i wins the pot with probability proportional to the index wiai for i with xi>0. The idea is that if player i bets more, then he is more likely to win, but this is modified by weights that parameterize the players’ abilities. Each player is trying to maximize his probability of taking all the chips (i.e., reaching xi=M). In the two-person game, it is known that a Nash equilibrium is for each player to adopt strategy σ of playing timidly (ai=1) or boldly (ai=xi) according to whether the game is in his favor or not (assuming the other also plays σ). In this article, we investigate whether this also is the form of a Nash equilibrium in a three-person game when the weights are of the form (w1, w2, w3)=(w, w, 1−2w) with 0 1/3 and M≥8.