Oliver Roy: Thinking Before Acting, pp. 1-50

I'm reading Oliver Roy's PhD thesis. So far, I've read the first 50 pages or so.

Roy uses a very classical picture of "instrumental reason" in which an agent's intentions are modelled as sets of outcomes. So, for instance, when I play rock-paper-scissors, I may "intend" different combinations of R, P, and S. In particular, I may "intend" losing combinations.

This is quite peculiar, but it also leads to a more versatile notion of rationalizability because players can wonder about not only the strategies of their fellow players, but also their intentions. (Such as they may not care about the outcome of the game or even play to lose.)

In chapter 2, he introduces a concept he calls "payoff-compatible intentions." This is a little convoluted and needs some unpacking:

He actually doesn't model intentions as sets of outcomes, but as a system of sets of outcomes, and he places a number of strong assumptions on those systems---essentially, he wants the system to be generated by a non-empty bottom element in the subset ordering. The intention system then coincide exactly with the system of supersets of this "most precise intention."

Using his assumptions, if you start from some intention set and then move upwards in the lattice that represents the subset ordering, you always stay inside the intention system. Being more lax about outcomes never disqualifies an intention. The other direction doesn't hold always, though: If you get more specific, you don't necessarily stay in the intention system.

A payoff-compatible intention system is then one in which you can move downwards if it doesn't lower your payoff. So, if A is a subset of B, and the outcomes in A are as good as the outcomes in B, then A will be in the intention set if B is.

In those cases, the bottom element in the intention system only consists of maximal elements with respect to the preference ordering (because smaller and better subsets are always also a part of the system). Payoff-compatible thus means very, very picky with respect to payoffs or preferences. In fact so picky that your most precise intention always only consists of the best possible outcomes. He uses this in chapter 3 to filter out Lo-Lo equilibria in Hi-Lo games.