In cooperative game theory he contributed to the solution concepts called the core and the Shapley value.
[13] In those set-ups Jean-François Mertens provided an extension of the characterization of the minmax and maxmin value for the infinite game in the dependent case with state independent signals.
Mertens also found the exact rate of convergence in the case of game with incomplete information on one side and general signalling structure.
'[22][23] Collectively Jean-François Mertens's contributions with Zamir (and also with Sorin) provide the foundation for a general theory for two person zero sum repeated games that encompasses stochastic and incomplete information aspects and where concepts of wide relevance are deployed as for example reputation, bounds on rational levels for the payoffs, but also tools like splitting lemma, signalling and approachability.
While in many ways Mertens's work here goes back to the von Neumann original roots of game theory with a zero-sum two person set up, vitality and innovations with wider application have been pervasive.
[24] The first paper studied the discounted two-person zero-sum stochastic game with finitely many states and actions and demonstrates the existence of a value and stationary optimal strategies.
Mertens clears orders creating a matching engine by using the competitive equilibrium – in spite of most usual interiority conditions being violated for the auxiliary linear economy.
This is the historical reason why some differentiability conditions have been originally required to define Shapley value of non-atomic cooperative games.
[29] This trick alone works well for majority games (represented by a step function applied on the percentage of population in the coalition).
Doing so, Mertens expends the diagonal formula to a much larger space of games, defining a Shapley value at the same time.
In particular since completely mixed Nash equilibrium are sequential – such equilibria when they exist satisfy both forward and backward induction.
In his work Mertens manages for the first time to select Nash equilibria that satisfy both forward and backward induction.
Elon Kohlberg and Mertens[34] emphasized that a solution concept should be consistent with an admissible decision rule.
Subsequently, Mertens[37][38] defined a refinement, also called stability and now often called a set of Mertens-stable equilibria, that has several desirable properties: For two-player games with perfect recall and generic payoffs, stability is equivalent to just three of these properties: a stable set uses only undominated strategies, includes a quasi-perfect equilibrium, and is immune to embedding in a larger game.
Essentiality requires further that no deformation of the projection maps to the boundary, which ensures that perturbations of the fixed point problem defining Nash equilibria have nearby solutions.
In a seminal paper Arrow (1950)[40] showed the famous "Impossibility Theorem", i.e. there does not exist an SWF that satisfies a very minimal system of axioms: Unrestricted Domain, Independence of Irrelevant Alternatives, the Pareto criterion and Non-dictatorship.
Relative Utilitarianism (RU) (Dhillon and Mertens, 1999)[41] is a SWF that consists of normalizing individual utilities between 0 and 1 and adding them, and is a "possibility" result that is derived from a system of axioms that are very close to Arrow's original ones but modified for the space of preferences over lotteries.
The theorem can be interpreted as providing an axiomatic foundation for the "right" interpersonal comparisons, a problem that has plagued social choice theory for a long time.
Relative utilitarianism[41] can serve to rationalize using 2% as an intergenerationally fair social discount rate for cost-benefit analysis.