A theorem of the maximin and applications to Bayesian zero-sum games

Abstract

Consider a family of zero-sum games indexed by a parameter that determines each player's payoff function and feasible strategies. Our first main result characterizes continuity assumptions on the payoffs and the constraint correspondence such that the equilibrium value and strategies depend continuously and upper hemicontinuously (respectively) on the parameter. This characterization uses two topologies in order to overcome a topological tension that arises when players' strategy sets are infinite-dimensional. Our second main result is an application to Bayesian zero-sum games in which each player's information is viewed as a parameter. We model each player's information as a sub-σ-field, so that it determines her feasible strategies: those that are measurable with respect to the player's information. We thereby characterize conditions under which the equilibrium value and strategies depend continuously and upper hemicontinuously (respectively) on each player's information. This clarifies and extends related results of Einy et al. (2008).

Power and Efficiency in Living Systems

“…To some scientists, the organismal trends support the view that "Evolution is the survival of the powerful" (p. 166 in [462]; also see Section 5.1 and [3,[8][9][10][11][12][13][139][140][141][142]480,481]). By contrast, other scientists have argued that evolution has increased the energetic or functional efficiency of organisms (also see Section 6 and [9,141,[246][247][248]482]).…”

“…Since all biological processes depend on metabolic energy, and natural selection favors reproductive success (gene transmission to the next generation), one might argue that natural selection should maximize the acquisition and conversion of resources (energy) into reproduction, i.e., "reproductive power" [3,[8][9][10][11][12][13]135,136]. The "struggle for life" described by Darwin [137] can be viewed as a "struggle for acquiring available energy" [138], and those organisms that do this best should outcompete and eventually replace relatively unsuccessful energy gatherers/users [3,9,[139][140][141].…”

“…The "struggle for life" described by Darwin [137] can be viewed as a "struggle for acquiring available energy" [138], and those organisms that do this best should outcompete and eventually replace relatively unsuccessful energy gatherers/users [3,9,[139][140][141]. Following this viewpoint, some biologists have suggested that the energetic power supporting reproduction (or, more broadly, somatic and reproductive production above maintenance costs) can be considered a useful non-tautological index of evolutionary fitness [10][11][12][13]142]. As such, it is a basic assumption of the "equal fitness paradigm", which posits that all species, small and large, have evolved to maximize their reproductive power relative to generation time, thus causing them all to have nearly equal energetic fitness, at least in a broad statistical sense [14][15][16] (but see Section 7.3.2).…”

“…All species strive to garner as much energy as they can, which often results in evolutionary "conflicts of self-interest" that inhibit the attainment of maximum power by any individual species. As a result, evolution is a "zero-sum game" [11], causing no species on Earth to usurp all the sun's energy for itself. As pointed out by Vermeij [139], since most resources are often other species, "acquisition and retention of resources by an individual organism requires that individual to prevent others from obtaining or monopolizing those resources" (p. 221).…”

“…Second, I focus on studies that use the concepts of power and efficiency as measures or indicators of evolutionary fitness or adaptation (Sections 5-7). Several biologists have suggested that maximal power (rate of resource use) for production (including specifically reproduction) is a useful energetic index of evolutionary fitness (Section 5 [3,7,[9][10][11][12][13]). Furthermore, J. H. Brown and colleagues [14] have argued that the universal action of natural selection favoring maximal power per generation has resulted in species of all sizes having essentially equivalent energetic fitness, a view called the "equal fitness paradigm" (also see [15,16]).…”