%0 Journal Article %1 akin1982exponential %A Akin, Ethan %D 1982 %J Canad. J. Math. %K evolutionary_stable_strategies game_theory superprocesses %N 2 %P 374--405 %R 10.4153/CJM-1982-025-4 %T Exponential families and game dynamics %U http://dx.doi.org/10.4153/CJM-1982-025-4 %V 34 %X mathreviews The author studies game dynamics in an ESS (evolutionary stable strategies) context. Suppose there are finitely many pure strategies. The simplex Δ whose vertices are the pure strategies is the space of all mixed strategies. If individuals in the population are restricted to pure strategies then the state of the population is described by a point x of Δ. The payoff matrix (aij), where i and j are pure strategies, provides a vector field on Δ which determines the evolution of x. This dynamic has been studied by the reviewer and L. B. Jonker Math. Biosci. 40 (1978), no. 1–2, 145–156; MR0489983 (58 #9351) and J. Hofbauer, P. Schuster and K. Sigmund J. Theoret. Biol. 81 (1979), no. 3, 609–12; MR0558663 (81d:92015). However, if individuals are permitted to use mixed strategies, the state of the population should really be described by a Borel probability measure on Δ. The dynamic in this case, on the space of all such measures, has been studied by W. G. S. Hines J. Appl. Probab. 17 (1980), no. 3, 600–610; MR0580020 (81i:92019) and E. C. Zeeman "Dynamics of the evolution of animal conflicts'', to appear. The present paper studies the relationship between these two dynamical systems. The author sees this work as "part of a growing bridge between differential geometry and statistics''. Does he mean probability theory?

@article{akin1982exponential, abstract = {[mathreviews] The author studies game dynamics in an ESS (evolutionary stable strategies) context. Suppose there are finitely many pure strategies. The simplex Δ whose vertices are the pure strategies is the space of all mixed strategies. If individuals in the population are restricted to pure strategies then the state of the population is described by a point x of Δ. The payoff matrix (aij), where i and j are pure strategies, provides a vector field on Δ which determines the evolution of x. This dynamic has been studied by the reviewer and L. B. Jonker [Math. Biosci. 40 (1978), no. 1–2, 145–156; MR0489983 (58 #9351)] and J. Hofbauer, P. Schuster and K. Sigmund [J. Theoret. Biol. 81 (1979), no. 3, 609–12; MR0558663 (81d:92015)]. However, if individuals are permitted to use mixed strategies, the state of the population should really be described by a Borel probability measure on Δ. The dynamic in this case, on the space of all such measures, has been studied by W. G. S. Hines [J. Appl. Probab. 17 (1980), no. 3, 600–610; MR0580020 (81i:92019)] and E. C. Zeeman ["Dynamics of the evolution of animal conflicts'', to appear]. The present paper studies the relationship between these two dynamical systems. The author sees this work as "part of a growing bridge between differential geometry and statistics''. {Does he mean probability theory?} }, added-at = {2015-03-06T16:58:14.000+0100}, author = {Akin, Ethan}, biburl = {https://www.bibsonomy.org/bibtex/2975d57b0b64147d71f7c4fbbe6278558/peter.ralph}, coden = {CJMAAB}, doi = {10.4153/CJM-1982-025-4}, fjournal = {Canadian Journal of Mathematics. Journal Canadien de Math\'ematiques}, interhash = {746be39d538ed366fe097681808e790a}, intrahash = {975d57b0b64147d71f7c4fbbe6278558}, issn = {0008-414X}, journal = {Canad. J. Math.}, keywords = {evolutionary_stable_strategies game_theory superprocesses}, mrclass = {92A15 (90D40)}, mrnumber = {658973 (83m:92052)}, mrreviewer = {Peter D. Taylor}, number = 2, pages = {374--405}, timestamp = {2015-03-06T16:58:14.000+0100}, title = {Exponential families and game dynamics}, url = {http://dx.doi.org/10.4153/CJM-1982-025-4}, volume = 34, year = 1982 }