A new solution concept for 2-person 0-sum games

[siddharth chatterjee]

Dear All,

Attached is a piece on a proposed (set-valued) solution concept for 2-person 0-sum games defined with preferences (rather than representations) as primitives.

The proposed solution concept is nonempty if players' opposite preferences satisfy (a weaker version of) vNM Independence though not necessarily Continuity. It generalises Nash/Minimax in the sense of including those if they exist, and coincides with those classical solution concepts if Continuity is assured in addition. The proposed solution concept, called consideration equilibria, enjoys interchangeability and indifference properties like minimax strategy tuples. The main idea is to define sets of admissible strategies and do iterated elimination akin to undominated strategies thereby leading to the final surviving sets.

The motivation for this work was Fishburn's paper (IJGT, 1971) presenting three examples of two-person games with opposite preferences, with Lexicographic Expected Utility representations, to illustrate the failure of classical solution concepts to dispel a myth which had become folk wisdom at a time in the appropriate circle that standard game theory should be possible with such preferences because maximization of such preferences over convex compact sets lead to nonempty convex compact sets. I analyse Fishburn's examples using the proposed solution concept.

I will not pretend that this idea is final or perfect in any sense. I look at it as one possible step towards the following question: What are some reasonable ways to play in general while not relying on technical assumptions like Continuity?

In case this interests you, I would be very happy.

Any feedback and comments on the present work, or for extensions and so on, would be greatly appreciated.

Kind regards,

Siddharth.

Postdoctoral Fellow, University of Essex

PhD, Indian Statistical Institute (Delhi Center)