Oliver Roy, the rest of the thesis (more or less)

There is a concept in Oliver Roy's thesis that I hadn't heard of before, and which intrigues me. It's the notion of Stackelberg solutions, which he briefly discusses in chapter 3.6.

A Stackelberg equilibrium is a strategy profile in which all players play as if their fellow players were mind readers; that is, everyone plays as if they were making the first move in an extensive game with perfect information. This naturally leads to Pareto efficient coordination in Hi-Lo games.

But what I find interesting about the concept is that Stackelberg equilibria (in dynamic systems) are to global maxima (in decision problems) as Nash equilibria are to local maxima.

To spell this out, think of game for e.g., 2 players, as a dynamic system consisting of two balls rolling around two inclined planes. The position of the first ball determines the inclination of the other plane and vice versa. You then find the Nash equilibria by looking for places to put the balls such that none of them are going to roll anywhere. Any such pair of positions will do, and we are thus content with any stationary points on the utility function (or its restriction to the boundary of the feasible set).

In a Stackelberg solution, we aren't content with any stationary point; we want them to be globally optimal. We are thus looking for the lowest stable position we can place one ball in such that the other one is also lying in its lowest stable position.

Such points may not exist. For instance, in Battle of the Sexes, there is no such equilibrium: If we place one ball in its lowest stable position, the other ball will lie a stable position, but not the lowest one.

In the Hi-Lo game he discusses, the Stackelberg outcome of playing Hi with probability x is a piecewise linear function:

• u(x) = 1 -- x    for 0 <= x < 1/3 (the other player plays Lo)
• u(x) = 2/3       for x = 1/3 (the other player plays whatever)
• u(x) = 2x        for 1/3 < x <= 1 (the other player plays Hi)

Its global maximum is thus obtained in x = 1 and has the value 2. This is the payoff associated with the perfect coordination in (Hi, Hi).

In Battle of the Sexes, the utilities look quite similar, but the globally optimal strategy of one player implies that the other player must play a strategy that isn't globally optimal.