Mixing times and moving targets

We consider irreducible Markov chains on a finite state space. We show that the mixing time of any such chain is equivalent to the maximum, over initial states $x$ and moving large sets $(A_s)_s$, of the hitting time of $(A_s)_s$ starting from $x$. We prove that in the case of the $d$-dimensional torus the maximum hitting time of moving targets is equal to the maximum hitting time of stationary targets. Nevertheless, we construct a transitive graph where these two quantities are not equal, resolving an open question of Aldous and Fill on a "cat and mouse" game

Cops vs. Gambler

We consider a variation of cop vs.\ robber on graph in which the robber is not restricted by the graph edges; instead, he picks a time-independent probability distribution on $V(G)$ and moves according to this fixed distribution. The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously. We show that when the gambler's distribution is known, the expected capture time (with best play) on any connected $n$-vertex graph is exactly $n$. We also give bounds on the (generally greater) expected capture time when the gambler's distribution is unknown to the cop.Comment: 6 pages, 0 figure

Contract Incentives and Effort

In a prevailing employment contract, the agent receives a proportional split of commissions. Alternatively, the agent receives a contract paying 100% of revenue above a fixed payment to the firm. In this contract the firm has a prior payment position, similar to a landlord or lender. The coexistence of these equity-only and debt-equity type contracts allows testing incentives for productivity and effort for U.S. real estate licensees. Hourly wages and productivity are increasing in the agent's split, up to and including 100%. Effort as measured by hours worked also positively affected by the split. The contract incentives motivate productivity and induce effort without requiring monitoring.