16/10/2024, 17:00 — 18:00 — Online
Alan Hammond, UC Berkeley
The Trail Of Lost Pennies: random turn games governed by stakes
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 will discuss the Trail of Lost Pennies, which is a tug-of-war game played on the integers in which the resources that each player expends during the game are deducted from her terminal receipt. The solution of this game is remarkably sensitive to the relative incentives of the two players, with a change in incentive of order $10^{-4}$ being sufficient to fundamentally alter outcomes. Overall we will indicate how stake-governed tug-of-war may weave the two strands of research in economics and mathematics, with the geometrically richer mathematical setting being allied to the resource-based rules for turn victor that is characteristic of the economics research strand.
