Convexity, Differential Equations, and Games

Theoretical and experimental studies of noncooperative games increasingly recognize Nash equilibrium as a limiting outcome of players‘ repeated interaction. This note, while sharing that view, illustrates and advocates combined use of convex optimization and differential equations, the purpose being to render equilibrium both plausible and stable.noncooperative games, Nash equilibrium, repeated play, differential equations, stability.

Balanced Environmental Games

Focus is here on coalitional games among economic agents plagued by aggregate pollutions of diverse sorts. Defecting players presumably pollute more than others. Then, granted convex preferences and technologies, the core is proven nonempty. In fact, under natural assumptions, a specific, computable core solution comes in terms of shadow prices on the said aggregates. Such prices may, in large part, implement the cooperative treaty by clearing a competitive market for emissions.Cooperative Games; Pollution Control Adoption Costs; Distribution Effects; Employment Effects

Social Insurance of Short Spell Sickness

This paper looks at social insurance of short term absence from work. The chief concern is with efficiency properties of full coverage. That arrangement is reviewed and criticized here in light of received theory. A main point is that positive loading of the premium implies less than full coverage. Concerns with optimal risk sharing also pull in the same direction. Besides, full coverage creates problems with moral hazard. The possibilities to self-insure over time are emphasized.risk sharing, coinsurance, deductible, non-insurable risk, Pareto efficiency, mutual insurance, arbitrage, adverse selection, moral hazard

Pooling, Pricing and Trading of Risks

Abstract. Exchange of risks is considered here as a transferableutility, cooperative game, featuring risk averse players. Like in competitive equilibrium, a core solution is determined by shadow prices on state-dependent claims. And like in finance, no risk can properly be priced only in terms of its marginal distribution. Pricing rather depends on the pooled risk and on the convolution of individual preferences. The paper elaborates on these features, placing emphasis on the role of prices and incompleteness. Some novelties come by bringing questions about existence, computation and uniqueness of solutions to revolve around standard Lagrangian duality. Especially outlined is how repeated bilateral trade may bring about a price-supported core allocation.Keywords: cooperative game; transferable utility; core; risks; mutual insurance; contingent prices; bilateral exchange; supergradients; stochastic approximation.

Full Coverage for Minor, Recurrent Losses?

This note looks at insurance of minor, recurrent losses. The main concern is with efficiency properties of full coverage. As motivation and running example we concider a regime, currently operative in several European countries, that offers employees complete wage reimbursement during short spell sickness. Assembled here are some arguments speaking against this sort of insurance policy.risk sharing; coinsurance; deductible; non-insurable risk; Pareto efficiency; mutual insurance; arbitrage; adverse selection; moral hazard.

Pooling, Pricing and Trading of Risks

Exchange of risks is considered here as a transferable-utility cooperative game. When the concerned agents are risk averse, there is a core imputation given by means of shadow prices on state-dependent claims. Like in finance, a risk can hardly be evaluated merely by its inherent statistical properties (in isolation from other risks). Rather, evaluation depends on the pooled risk and the convolution of individual preferences. Explored below are relations to finance with some emphasis on incompleteness. Included is a process of bilateral trade which converges to a price-supported core allocation.

Stochastic Approximation, Momentum, and Nash Play

Main objects here are normal-form games, featuring uncertainty and noncooperative players who entertain local visions, form local approximations, and hesitate in making large, swift adjustments. For the purpose of reaching Nash equilibrium, or learning such play, we advocate and illustrate an algorithm that combines stochastic gradient projection with the heavyball method. What emerges is a coupled, constrained, second-order stochastic process. Some friction feeds into and stabilizes myopic approximations. Convergence to Nash play obtains under seemingly weak and natural conditions, an important one being that accumulated marginal payoffs remains bounded above.Noncooperative games; Nash equilibrium; stochastic programming and approximation; the heavy ball method.

Computing Normalized Equilibria in Convex-Concave Games

Abstract. This paper considers a fairly large class of noncooperative games in which strategies are jointly constrained. When what is called the Ky Fan or Nikaidô-Isoda function is convex-concave, selected Nash equilibria correspond to diagonal saddle points of that function. This feature is exploited to design computational algorithms for finding such equilibria. To comply with some freedom of individual choice the algorithms developed here are fairly decentralized. However, since coupling constraints must be enforced, repeated coordination is needed while underway towards equilibrium. Particular instances include zero-sum, two-person games - or minimax problems - that are convex-concave and involve convex coupling constraints.Noncooperative games; Nash equilibrium; joint constraints; quasivariational inequalities; exact penalty; subgradient projection; proximal point algorithm; partial regularization; saddle points; Ky Fan or Nikaidô-Isoda functions.

Investment Uncertainty, and Production Games

This paper explores some cooperative aspects of investments in uncertain, real options. Key production factors are assumed transferable. They may reflect property or user rights. Emission of pollutants and harvest of renewable resources are cases in point. Of particular interest are alternative projects or technologies that provide inferior but anti-correlated returns. Any such project stabilizes the aggregate proceeds. Therefore, given widespread risk exposure and aversion, that project’s worth may embody an extra bonus. The setting is formalized as a stochastic production game. Granted no economies of scale such games are quite tractable in analysis, computation, and realization. A core imputation comes in terms of contingent shadow prices that equilibrate competitive, endogenous markets. The said prices emerge as optimal dual solutions to coordinated production programs, featuring pooled resources - and also via adaptive procedures. Extra value - or an insurance premium - adds to any project whose yield is negatively associated with the aggregate.investment, risk attitudes, insurance, covariance-pricing, cooperative games, core, stochastic optimization

Convexity, convolution and competitive equilibrium

This paper considers a chief interface between mathematical programming and economics, namely: money-based trade of perfectly divisible and transferable goods. Three important and related features are singled out here: first, convexity enters via acceptable payments, second, convolution of monetary criteria secures Pareto efficiency, and third, competitive equilibrium obtains when agents' subdifferentials intersect.publishedVersio