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The principle of monotonicity for cooperative games states that if a game changes so that some player's contribution to all coalitions increases or stays the same then the player's allocation should not decrease. There is a unique symmetric and efficient solution concept that is monotonic in this most general sense -the Shapley value. Monotonicity thus provides a simple characterization of the value without resorting to the usual "additivity" and "dummy" assumptions, and lends support to the use of the value in applications where the underlying "game" is changing, e.g. in cost allocation problems.Monotonicity is a general principle of fair division which states that as the underlying data of a problem change, the solution should change in parallel fashion. It is particularly germane to applications in which allocations are not made once and for all, but are reassessed periodicaUy as new information emerges. This is the case, for example, in dividing the joint benefits or costs of a cooperative enterprise fairly among the partners when the underlying structure of the enterprise is evolving over time. Such a situation can be modelled by a cooperative game.We give several formulations of the monotonicity principle for cooperative games and characterize different solution concepts v/a this property. Some commonly advocated methods -including the nucleolus and the so-called separable costs remaining benefits method -fail even the weakest test of monotonicity. In a somewhat more comprehensive form, monotonicity is shown to be inconsistent with a solution staying in the core. Finally, in a still stronger form it is shown to be consistent with exactly one (symmetric and efficient) solution concept -the Shapley value.A cooperative game with players {1,2 .... , n) = N is a real valued function v (S) defined on all coalitionsS _CN such that v (r = 0. v (S) is the value of S. Although we do not require it, v is often assumed to be superadditive

New ideas, products, and practices take time to diffuse, a fact that is often attributed to some form of heterogeneity among potential adopters. This paper examines three broad classes of diffusion models -- contagion, social influence, and social learning -- and shows how to incorporate heterogeneity into each at a high level of generality without losing analytical tractability. Each type of model leaves a characteristic "footprint" on the shape of the adoption curve which provides a basis for discriminating empirically between them. The approach is illustrated using the classic study of Ryan and Gross (1943) on the diffusion of hybrid corn. (JEL D83, O33, Q16, Z13)

Interconnections among financial institutions create potential channels for contagion and amplification of shocks to the financial system. We estimate the extent to which interconnections increase expected losses, with minimal information about network topology, under a wide range of shock distributions. Expected losses from network effects are small without substantial heterogeneity in bank sizes and a high degree of reliance on interbank funding. They are also small unless shocks are magnified by some mechanism beyond simple spillover effects; these include bankruptcy costs, fire sales, and mark-to-market revaluations of assets. We illustrate the results with data on the European banking system.