Hello again. Now that finals and break are over I can return to my haphazard updating schedule. No, I don't know what happened to January either.
Today's post isn't about a particular game, but rather an assumption that is important to understanding game theory. When a game theoretic model gets described, we assume that the players are rational actors. However, rationality in this sense might be different from a more conventional understanding of the word. Most common (or at least, the first one that comes to mind), is some reference to a person forming conclusions logically and acting accordingly. We might also assume that a rational person would make good choices, or at least sensible ones. Having seen a few people walking outside in the Rochester winter in shorts, it's obvious that this is not how all people behave. So what's an economist/political scientist to do?
To answer that question, let's take a minor digression into utility functions (my favorite functions!). A utility function is an assignment of utility values to alternatives. Please contain your enthusiasm, it gets better. Anyway, a utility value is just a number, and we know what alternatives are. If you have to pick between beer, quiche, or a fight, you might assign 1 to beer, 2 to quiche, and 3 to a fight. That's a utility function. It doesn't exactly matter what number you assign to what, so long as bigger numbers mean better alternatives. Neither do the relative magnitudes of utility values matter. For example, you might assign -1,000,000 to beer, 3.4 to quiche, and 500 to a fight. This would be no different from the other function. Technically, utility functions of this form are called ordinal utility functions. If this seems strange, have patience! I'll try and explain EVEN MORE.
An ordinal utility function therefore allows us to represent a ranking among alternatives. In game theory, this is critically important. In fact, the game theoretic assumption of rationality simply requires that a person be able to rank alternatives such that the top ranked one is what they most prefer. Utility functions then are devices that represent those preferences so a game can be made solvable. So, looking back at the person outside in shorts, it may be that they prefer shorts to pants no matter what. This could be perfectly rational behavior, theoretically speaking, since we don't make assumptions about the quality of the ranking a person creates.
Maybe this should actually be categorized under introductory material. But this assumes that I am able to create comprehensible organization schemes, a contradiction.
