EC 490 Week 2

1. `Games’ in the everyday sense involve three elements in different proportions:

- chance (pure luck)

- skill (physical or mental) and knowledge (of facts or of logic and mathematics)

- strategy (calculated best use of skill)

`Games’ in the technical sense of GT aren’t those involving pure chance or pure skill. Games in the technical sense are strategic: interactions between mutually aware players.

2. Contrast games (non-parametric situations) with decisions - parametric situations where each agent can choose without concern for reaction or response from others.

3. Dimensions along which games vary:

- Sequential or simultaneous? Does each player know, when she chooses her moves, what other players have done up to that point?

- Zero-sum (constant sum) or are interests partially aligned?

- Repeated or one-shot? Do players have reasons to be concerned about the effects of their strategies on their reputations?

- Asymmetric (incomplete) or symmetric (complete) information? Does one player have information the others lack? If so, does she want them to know it or not? If no, can she take actions to conceal it? Can other players take screening actions that will force her to reveal it? If a player wants to signal her private information – thus implying that she gets a strategic advantage from other players believing this information to be true – how can she do so in a way that will convince other players she isn’t lying?

- Cooperative or non-cooperative? This is a technical distinction which you can ignore in this course. It is strictly a course in non-cooperative game theory.

Elements of game theory

4. Each player in a game faces a choice among two or more possible strategies. A strategy is a predetermined complete ‘programme of play’ that tells her what actions to take in response to every possible strategy other players might use. The significance of the italicized phrase here will become clear when we take up some sample games shoertly. A strategy can sometimes tell a player to flip a (weighted or unweighted) coin at certain points in a game. In that case we say the strategy is mixed.

5. The next concept we need is that of a utility function. Remember that an agent is, by definition, an entity with preferences. Economists describe these by means of an abstract concept called utility. This refers to how an agent ranks an object or event, so that an agent who, for example, adores the taste of pickles would be said to associate high utility with them, while an agent who can take or leave them associates a lower level of utility with them. Examples of this kind suggest that ‘utility’ denotes a measure of subjective psychological fulfillment, and this is indeed how the concept was generally (though not always) interpreted prior to the 1930s. During that decade, however, economists rejected the theoretical use of such unobservable entities as ‘psychological fulfillment quotients.’ (Psychologists have since discovered that there are in fact no such things.) They therefore defined utility in such a way that it became a purely technical concept. When we say that an agent acts so as to maximize her utility, we mean by ‘utility’ simply whatever it is that the agent's behavior suggests she is consistently pursuing. This could refer to material goods, the happiness of others, world peace, tantric bliss, or anything.

6. Since game theory involves formal reasoning, we must have a device for thinking of utility maximization in mathematical terms. Such a device is called a utility function. The utility-map for an agent is called a ‘function’ because it maps ordered preferences onto the real numbers. Suppose we have three bundles over which agent X has preferences. (A `bundle’ might be: a new Toyota, a house in Hoover, good sex four times a week and the establishment of stable democracy in Iraq this year. Or anything else.) Suppose that X prefers bundle a to bundle b and bundle b to bundle c. We then map these onto a list of numbers, where the function maps the highest-ranked bundle onto the largest number in the list, the second-highest-ranked bundle onto the next-largest number in the list, and so on, thus:

bundle a ® 3

bundle b ® 2

bundle c ® 1

In the games we’ll study at first, the only property mapped by this function is order. So such functions are called ordinal utility functions. The magnitudes of the numbers are irrelevant; that is, you must not read the function above as saying that X gets 3 times as much utility from bundle a as she gets from bundle c. Thus we could represent exactly the same ordinal utility function as that above by

bundle a ® 7,326

bundle b ® 12.6

bundle c ® -1,000,000

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. Later in the course, when we come to seeing how to solve games that involve mixed strategies we'll need to treat utility functions as cardinal. The technique for doing this was given by von Neumann & Morgenstern, and was an essential aspect of their invention of game theory. It is what Dixit and Skeath are referring to when they talk about expected utility. For the moment, however, we will need only ordinal utility functions.

7. When numbers drawn from utility are assigned to the conclusions of games they’re called payoffs. An outcome of a game must specify a payoff for every player in the game, drawn in each case from the respective player’s utility function.

8. Rationality: We assume that players can (i) assess outcomes; (ii) calculate paths to outcomes; and (iii) choose actions that yield their most-preferred outcomes, given the actions of the other players.

9. `Rational’ in economics and game theory does not mean anything else. In particular, it doesn’t mean: (i) moral; (ii) sensible; (iii) self-conscious about one’s plans and reasons.

10. An agent is any rational entity. So the following are all examples of agents: (i) a (sane) person; (ii) a firm; (iii) a non-human animal; (iv) a plant; (v) a labour union; (vi) a brain cell … and many others. Examples of non-agents: rocks, cars, buildings, planets, houses. Any agent can be a player in a game. No non-agent can be.
11. Are agents really rational? They only approximate rationality in situations that aren’t novel to them. Non-human animals are much more rational than people, because they face fewer novel situations for which evolution didn’t prepare them. However, there’s a good reason for modeling people as rational for purposes of designing policies. You want your response to the strategies of others to continue to work as the others learn improved performance. This is especially clear in business contexts. Is Coke better off modeling Pepsi as smarter than it really is, or as cleverer?

12. We assume until much later in the course that all players in a game know its rules. These consist of: (i) the list of players; (ii) the strategies available to each player (the structure of the game); (iii) the utility functions of all players; (iv) the assumption that all players are rational.

13. Solving a game consists in finding its equilibria (plural of equilibrium). We will postpone consideration of this for now.

14. It is possible to drop the assumption that players know the rules of their game. One then turns to evolutionary game theory, in which players learn equilibria by trial and error. We will study how this works toward the end of the course. Most of the games in which people are involved are evolutionary games. They generally play such games without realizing they’re doing so.

15. But let’s not try to run before we can walk. We’ll begin next week with the simplest possible games: sequential games in which a small number of players who perfectly know the rules of their games choose from amongst small sets of available strategies and don’t mix.