Notes on Testing Game Theory

This paper is a bit philosophical, I would say. It provides some criticism of experimental studies, which seem to be a fad these days. I enjoyed reading it for two reasons. First, it talked about some questions I had when I took Behavioral Economics I. The key message to me is that game theory is not a consequentialist theory. As Jörgen stressed in class, we need to be more careful about setting up plays and assuming preferences. Many experiments seem to say: look at my experiment, people do not play Nash equilibrium!!! But this argument may not be valid since we cannot say what the equilibrium is without carefully identifying the preferences. Jörgen suggested testing joint hypothesis even epistemic modes of games cannot be falsified. He also mentioned (in class) that we should be more careful when talking about fairness...(I really like this point.)

Game theory was first developed by mathematicians. I guess they do not care about these subtleties. Economics is a science of human beings anyhow and preferences become very very important in characterizing a problem.

Secondly, it represents basic terminologies of game theory clearly, in words instead of math.



I. Basic terminologies


Game: (N, A, fai, P, I, C, p, r, v)

Game Form: (N, A, fai, P, I, C, p)

Game Protocol: (N, A, fai, P, I, C, p, r)

N: the set of personal players.

Such a game is a mathematical object that contains as its basic structure a directed tree, consisting of a finite number of nodes (or vertices) and branches (or edges). A play, tao, of the game is a route through the tree, starting at its initial node and ending at one of the end nodes, w. A node k’ is a successor of a node k if there is play that leads first to k and then to k’. Moreover, each end node is reached by exactly one play of the game, and each play reaches exactly one end node.

…

The set of non-end nodes is partitioned into player subsets, and each player subset in turn is partitioned into information sets for the player role. In each information set, the number of outgoing branches from each node is the same, and the set of outgoing branches from an information set is divided into equivalence classes, the moves available to the player at that information, so that every equivalence class contains exactly one outgoing branch from each node in the information set. A choice in an information set is a probability distribution over the moves available at the information set. In the game with exogenous random moves, one of players is ‘nature’, and all information sets for this non-personal player are singleton sets with fixed probabilities attached to each outgoing branch.


Pure Strategy/outcome

A pure strategy for a personal player role is a function that assigns one move to each of the role’s information sets. The outcome of a strategy profiles is the probability distribution induced on the set of end-nodes, or on the set of plays.

Subform

In a given game form Ksai, let K_0 be the subset of nodes k such that (i) k is either a move by nature or {k} is an information set of a personal player; and (ii) no information set in Ksai contains both a successor node and a non-successor node to k. Each node k in K_0 is the initial node of a subform.

(Here Jörgen talked about the subtlety of defining a subgame, i.e. whether it is context-related or isolated at k. It seems to me that this argument shares the same spirit of non- consequentialism. See Binmore et al (2002).)


II Levine's theory and discussion

Is the linear function a super clever idea or an arbitrary assumption? Is it robust?

Can I think of a better option?:)