The numbers featuring in an ordinal utility function are thus not measuring any quantity of anything. A utility-function in which magnitudes do matter is called ‘cardinal’

It was said above that games of perfect information are the (logically) simplest sorts of games. This is so because in such games (as long as the games are finite, that is, terminate after a known number of actions) players and analysts can use a straightforward procedure for predicting outcomes. A rational player in such a game chooses her first action by considering each series of responses and counter-responses that will result from each action open to her. She then asks herself which of the available final outcomes brings her the HIGHEST UTILITY, and chooses the action that starts the chain leading to this outcome. This process is called BACKWARD INDUCTION (because the reasoning works backwards from eventual outcomes to present decision problems).

Trees are used to represent sequential games, because they show the order in which actions are taken by the players. However, games are sometimes represented on matrices rather than trees. This is the second type of mathematical object used to represent games. Matrices, unlike trees, simply show the outcomes, represented in terms of the players' utility functions, for every possible combination of strategies the players might use. For example, it makes sense to display the river-crossing game from Section 1 on a matrix, since in that game both the fugitive and the hunter have just one move each, and each chooses their move in ignorance of what the other has decided to do.

Prisoner's Dilemma:
Wherever one action for a player is superior to her other actions for each possible action by the opponent, we say that the first action strictly dominates the second

When people introduce the PD into popular discussions, you will sometimes hear them say that the police inspector must lock his prisoners into separate rooms so that they can't communicate with one another. The reasoning behind this idea seems obvious: if you could communicate, you'd surely see that you're both better off refusing, and could make an agreement to do so, no? This, one presumes, would remove your conviction that you must confess because you'll otherwise be sold up the river by your partner. In fact, however, this intuition is misleading and its conclusion is false.

When we represent the PD as a strategic-form game, we implicitly assume that the prisoners can't attempt collusive agreement since they choose their actions simultaneously. In this case, agreement before the fact can't help. If you are convinced that your partner will stick to the bargain then you can seize the opportunity to go scot-free by confessing. Of course, you realize that the same temptation will occur to her; but in that case you again want to make sure you confess, as this is your only means of avoiding your worst outcome. Your agreement comes to naught because you have no way of enforcing it; it constitutes what game theorists call ‘cheap talk’.

[My experience with Kiw Seng: Even if me and mervyn were able to communicate before we sent for questioning, the results may be unconfirmed still. We both agree that we should choose to refuse to confess since it brings the best for both of us. However, given that i know that mervyn is sure to confess, i may choose not to do so. my decision to confess or not depends on the difference in value i place with 1. the extra punishment i would be saved from if i confessed and he didnt, 2. the strength of the friendship i place with mervyn and fear of any backlash.]

Node: A point at which a player takes an action.
Initial node: The point at which the first action in the game occurs.
Terminal node: Any node which, if reached, ends the game. Each terminal node corresponds to an outcome.
Subgame: Any set of nodes and branches descending uniquely from one node.
Payoff: an ordinal utility number assigned to a player at an outcome.
Outcome: an assignment of a set of payoffs, one to each player in the game.
Strategy: a program instructing a player which action to take at every node in the tree where she could possibly be called on to make a choice


... We can put this another way: in a zero-sum game, my playing a strategy that maximizes my minimum payoff if you play the best you can, and your simultaneously doing the same thing, is just equivalent to our both playing our best strategies, so this pair of so-called ‘maximin’ procedures is guaranteed to find the unique solution to the game, which is its unique NE. (In tic-tac-toe, this is a draw. You can't do any better than drawing, and neither can I, if both of us are trying to win and trying not to lose.)

[trying to win and preventing from losing is two different things]

posted by iambrianfu [ 12:48 PM ] <$BlogItemComments$>

Part of the explanation for game theory's relatively late entry into the field lies in the problems with which economists had historically been concerned. Classical economists, such as Adam Smith and David Ricardo, were mainly interested in the question of how agents in very large markets -- whole nations -- could interact so as to bring about maximum monetary wealth for themselves. Smith's basic insight, that efficiency is best maximized by agents freely seeking mutually advantageous bargains, was mathematically verified in the twentieth century. However, the demonstration of this fact applies only in conditions of ‘perfect competition,’ that is, when firms face no costs of entry or exit into markets, when there are no economies of scale, and when no agents' actions have unintended side-effects on other agents' well-being. Economists always recognized that this set of assumptions is purely an idealization for purposes of analysis, not a possible state of affairs anyone could try (or should want to try) to attain. But until the mathematics of game theory matured near the end of the 1970s, economists had to hope that the more closely a market approximates perfect competition, the more efficient it will be. No such hope, however, can be mathematically or logically justified in general; indeed, as a strict generalization the assumption can be shown to be false.
This article is not about the foundations of economics, but it is important for understanding the origins and scope of game theory to know that perfectly competitive markets have built into them a feature that renders them susceptible to parametric analysis. Because agents face no entry costs to markets, they will open shop in any given market until competition drives all profits to zero. This implies that if costs and demand are fixed, then agents have no options about how much to produce if they are trying to maximize the differences between their costs and their revenues. These production levels can be determined separately for each agent, so none need pay attention to what the others are doing; each agent treats her counterparts as passive features of the environment. The other kind of situation to which classical economic analysis can be applied without recourse to game theory is that of monopoly. Here, quite obviously, non-parametric considerations drop out, since there is only one agent under study. However, both perfect and monopolistic competition are very special and unusual market arrangements. Prior to the advent of game theory, therefore, economists were severely limited in the class of circumstances to which they could neatly apply their models.
Philosophers share with economists a professional interest in the conditions and techniques for the maximization of human welfare. In addition, philosophers have a special concern with the logical justification of actions, and often actions must be justified by reference to their expected outcomes. Without game theory, both of these problems resist analysis wherever non-parametric aspects are relevant. We will demonstrate this shortly by reference to the most famous (though not the most typical) game, the so-called Prisoner's Dilemma, and to other, more typical, games. In doing this, we will need to introduce, define and illustrate the basic elements and techniques of game theory. To this job we therefore now turn.

posted by iambrianfu [ 12:38 PM ] <$BlogItemComments$>