Stake-governed random-turn games
In 1987, Harris and Vickers [HV87] proposed a model of a race in which two firms invest resources, each trying to be the first to secure a patent. They called the model tug of war. A counter moves randomly left or right on an integer interval, with the odds of a rightward move at each turn determined by the resources expended by each of the firms at the turn. The game ends when the counter reaches one or another end of the interval, the patent thus accorded to one or another firm. In 2009, Peres, Schramm, Sheffield and Wilson [PSSW09] independently introduced a similar game, which they also named tug of war. Two players also push a counter on a board, with a trivial rule for turn victor selection, via the toss of a fair coin; but also with a much richer geometric setting: by considering the game played in the limit of small step size in a domain in Euclidean space, the value of the game was found to be infinity harmonic, namely to satisfy an infinity version of the usual Laplace equation. These two works, HV87 and PSSW09, have unleashed two big but thus far non-interacting waves of attention, from economists and mathematicians respectively. In this talk, we discuss recent joint work with Gábor Pete on stake-governed tug-of-war games, which were inspired by PSSW around 2004. In such a game, two players are each allocated a fixed budget at the start, from which they must draw funds throughout the game's lifetime in an effort to exert ongoing influence on the random movement of the counter on the board (and thus on its terminal location). The game's solution on certain graphs develops the connection of tug of war with infinity harmonic functions that PSSW09 identified while providing a rigorous basis for aspects of the considerable economics literature that HV87 inspired. Although we were until recently quite unaware of the history of tug of war in economics, the solution of certain forms of stake-governed tug-of-war seems to present a point of convergence of the mathematical and economic research efforts, providing a detailed rigorous solution in a geometrically richer context than is typically found in economics, and introducing to mathematicians a form of tug of war with a resource-based rule for turn victor that is characteristic of the economics research strand.