Ultimatum Game

I recently have begun collecting and reading journal articles in various fields related to what I plan to do research on. I have been looking for articles relating to the relatively new field of neuroeconomics and the search has actually not proved as easy as I thought. The real problem lies with the majority of the material I have been finding is in psychology journals and is lacking in mathematical content. However, these papers do provide some insight into that area of work and definetly helps me understand some of the current problems being pursued. The problem that I have before me is to find a direction for research that involves my interest in neuroscience and economics, but that also is significantly mathematical in nature. This should not be incredibly difficult to find and I believe that some of the papers I have been reading, although nonmathematical, will help me to this end.

A 2003 paper in Science titled "The neural basis of economic decision-making in the Ultimatum Game" by Sanfey, et. al. was quite fascinating to me. They essentially seem to provide empirical evidence for irrational behavior. Therefore, rationality assumptions in economic models may not be accurate in predicting behavior in the real world. Traditionally, the joke is that economic models tend to always make assumptions that never hold in reality. Furthermore, the real difficulty in my mind seems to be in how to best implement the relaxation of rationality.

A specific example from this paper might be the best way to explain. The Ultimatum game is one where a fixed amount of money is to be split among two players. Player A makes an offer to Player B about how to split the money. Player B may accept the offer and each receives their respective payoffs, or Player B may reject and both receive nothing. The game is played only once and after Player B's decision the game is over. Rational players we traditional would expect to play such that Player A offers as little as possible to Player B and Player B will always accept as long as they are offered a positive quantity of money. This is because we assume that rational people prefer any amount of money to none. But, when a deal is viewed as "too unfair", the experimentors found that the second player rejected offers. Neurologically this was correlated to increased activity in the bilateral anterior insula. This area of the brain is often associated with negative emotional responses. Therefore, they essentially found that negative emotions in the brain can outweigh cognitive decision-making areas of the brain and cause one to act irrationally. Player B consciously turns money down in order to spite Player A whom they believe made an unfair offer. How do we reconcile an economic model to such a result?

It seems that the area that needs to be researched is a better definition of game theoretic equilibrium. We need to somehow add to models of decision behavior the fact that emotions can alter decisions from pure rationality. To do so seems almost impossible though just at a glace. However, that is the purpose of doing something original and what will take time and effort. Possibly one can introduce a rationality function that is specific to the decision at hand and that is responsive to how "fair" or "unfair" a situation might be. Further, the situation is completely different if a game like the ultimatum game is played repeatedly.

My simple yet original idea can be explicated by seeing how one might examine the payoff structure of the ultimatum game with irrationality. Let the total amount of money to be split be denoted by $T. Then the payoff to player A is defined as: P(A) = f(x)*x; where x is the amount that player A chooses to give herself in the deal. We then define y = T - x, and see that this is the amount offered to player B in the proposed deal. Therefore, P(B) = f(x)*y. We are simply left to define f(x). We want to have a function that is equal to 1 when the deal is accepted by player B and equal to 0 when the deal is rejected. This function is thus a rationality function. If we assume pure rationality then f(x) = 1 for all x. However, we can introduce an infinite amount of possible irrational behaviors by player B that will define the ultimate payout. For example, let us define f(x) = 1 for x <= 7 and f(x) = 0 for x > 7. This would follow along the same lines as the experiment in the aforementioned paper where the majority of deals where the money was to be split 8/2 were rejected and the 7/3 deals were accepted. We could further introduce a stochastic component and use a dirac delta function multipled by a separate continuous function in order to more accurately create an f(x) that models experimental behavior.

I believe that something along these lines, obviously more well developed, could be used along with different notions of equilibrium to create more accurate economic models. We could, for example, attempt to find robust equilibrium conditions that hold even when irrationality is assumed. This would be a giant step foward for macroeconomic theory where we tend to always have to assume a rational reprsentative consumer in order to aggregate much of the strong results from microeconomics.