This paper endogenizes the frequency of major discoveries and the extent of their refinement.Four axioms deliver a one-parameter family of beliefs that guide exploratory effort.Such effort trades off the prospect of major new discovery against the chance of successfully refining discoveries made in the past.The only other parameter is the cost of making new discoveries relative to the cost of refining old ones.The paper derives time-series properties of inventive activity as they relate to the two parameters, and it discusses several specific inventions and their subsequent refinement.In doing so, the paper arguably enhances our understanding of the process of discovery.1 We thank the C. V. Starr Center for Applied Economics for technical and financial assistance.The second

We analyze an evolutionary model with a finite number of players and with noise or mutations.The expansion and contraction of strategies is linked-as usual-to their current relative success, but mutations-which perturb the system away from its deterministic evolution-are present as well.Mutations can occur in every period, so the focus is on the implications of ongoing mutations, not a one-shot mutation.The effect of these mutations is to drastically reduce the set of equilibria to what we term "long-run equilibria."For 2 x 2 symmetric games with two symmetric strict Nash equilibria the equilibrium selected satisfies (for large populations) Harsanyi and Selten's (1988) criterion of risk-dominance.In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected, even though there is another strict Nash equilibrium.

This paper elucidates the logic behind recent papers which show that a unique equilibrium is selected in the presence of higher order uncertainty, i.e., when players lack common knowledge.We introduce two new concepts: belief potential of the information system and p-dominance of Nash-equilibria of the game, and show that a Nash-equilibrium is uniquely selected whenever its p-dominance is below the belief potential.This criterion applies to many-action games, not merely 2 x 2 games.It also applies to games without dominant strategies, where the set of equilibria is shown to be smaller and simpler than might be initially conjectured.Finally, the new concepts help understand the circumstances under which the set of equilibria varies with the amount of common knowledge among players.