# American Institute of Mathematical Sciences

July  2022, 9(3): 299-322. doi: 10.3934/jdg.2022015

## On two-player games with pure strategies on intervals $[a, \; b]$ and comparisons with the two-player, two-strategy matrix case

 City, University of London, UK

*Corresponding author: Zahra Gambarova

Received  November 2021 Revised  April 2022 Published  July 2022 Early access  June 2022

We consider games of two-players with utility functions which are not necessarily linear on the product of convex and compact intervals of $\mathcal{R}^2$. An issue is how far an analogy can be drawn with two-player, two-strategy matrix games with linear utility functions, where [0, 1] registers probabilities and equilibria are at the intersection of reaction functions. Now, the idea of $\delta$ functions is exploited to construct mixed strategies to look for Nash equilibria (NE). "Reaction" functions are constructed and results are obtained graphically. They are related to topological theorems on NE. The games chosen make specific points in relation to existence conditions and properties of solutions. It is a distinguishing feature that an interval [a, b] now registers both pure and mixed strategies. For NE a choice has to be justified. Also "reaction" functions are more complicated and their intersection does not guarantee an equilibrium.

Citation: Zahra Gambarova, Dionysius Glycopantis. On two-player games with pure strategies on intervals $[a, \; b]$ and comparisons with the two-player, two-strategy matrix case. Journal of Dynamics and Games, 2022, 9 (3) : 299-322. doi: 10.3934/jdg.2022015
##### References:
 [1] C. D. Aliprantis and S. K. Chakrabarti, Games and Decision Making, 2$^{nd}$ edition, Oxford University Press, 2011. [2] C. D. Aliprantis, D. Glycopantis and D. Puzello, The joint continuity of the expected payoff functions, J. Math. Econom, 42 (2006), 121-130.  doi: 10.1016/j.jmateco.2005.06.002. [3] K. Binmore, Playing for Real; A Text on Game Theory, 2$^{nd}$ edition, Oxford University Press, 2007.  doi: 10.1093/acprof:oso/9780195300574.001.0001. [4] G. Debreu, Social equilibrium existence theorem, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 886-893.  doi: 10.1073/pnas.38.10.886. [5] K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121. [6] I. L. Glicksberg, A further generalization of the kakutani fixed point theorem, with application to nash equilibrium points, Proc. Amer. Math. Soc., 3 (1952), 170-174.  doi: 10.2307/2032478. [7] D. Glycopantis, On games with imperfect recall and games with perfect recall, SSRN-id2520455, (2014). [8] P. Morris, Introduction to Game Theory, Springer, 1994. [9] J. F. Nash, Equilibrium points in n-person games, Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48. [10] J. F. Nash, Non-cooperative games, Ann. of Math., 54 (1951), 286-295.  doi: 10.2307/1969529. [11] P. J. Reny, Non-cooperative Games (Equilibrium Existence), Palgrave Macmillan (eds) The New Palgrave Dictionary of Economics, Palgrave Macmillan, London, 2008. [12] F. Vega-Redondo, Economics and the Theory of Games, Cambridge University Press, Cambridge, 2022.

show all references

##### References:
 [1] C. D. Aliprantis and S. K. Chakrabarti, Games and Decision Making, 2$^{nd}$ edition, Oxford University Press, 2011. [2] C. D. Aliprantis, D. Glycopantis and D. Puzello, The joint continuity of the expected payoff functions, J. Math. Econom, 42 (2006), 121-130.  doi: 10.1016/j.jmateco.2005.06.002. [3] K. Binmore, Playing for Real; A Text on Game Theory, 2$^{nd}$ edition, Oxford University Press, 2007.  doi: 10.1093/acprof:oso/9780195300574.001.0001. [4] G. Debreu, Social equilibrium existence theorem, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 886-893.  doi: 10.1073/pnas.38.10.886. [5] K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121. [6] I. L. Glicksberg, A further generalization of the kakutani fixed point theorem, with application to nash equilibrium points, Proc. Amer. Math. Soc., 3 (1952), 170-174.  doi: 10.2307/2032478. [7] D. Glycopantis, On games with imperfect recall and games with perfect recall, SSRN-id2520455, (2014). [8] P. Morris, Introduction to Game Theory, Springer, 1994. [9] J. F. Nash, Equilibrium points in n-person games, Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48. [10] J. F. Nash, Non-cooperative games, Ann. of Math., 54 (1951), 286-295.  doi: 10.2307/1969529. [11] P. J. Reny, Non-cooperative Games (Equilibrium Existence), Palgrave Macmillan (eds) The New Palgrave Dictionary of Economics, Palgrave Macmillan, London, 2008. [12] F. Vega-Redondo, Economics and the Theory of Games, Cambridge University Press, Cambridge, 2022.
 [1] Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics and Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537 [2] Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091 [3] Athanasios Kehagias. A note on the Nash equilibria of some multi-player reachability/safety games. Journal of Dynamics and Games, 2022, 9 (1) : 117-122. doi: 10.3934/jdg.2021028 [4] Abderrahmane Habbal, Moez Kallel, Marwa Ouni. Nash strategies for the inverse inclusion Cauchy-Stokes problem. Inverse Problems and Imaging, 2019, 13 (4) : 827-862. doi: 10.3934/ipi.2019038 [5] Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics and Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 [6] Alejandra Fonseca-Morales, Onésimo Hernández-Lerma. A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models†. Journal of Dynamics and Games, 2017, 4 (3) : 195-203. doi: 10.3934/jdg.2017012 [7] Margarida Carvalho, João Pedro Pedroso, João Saraiva. Electricity day-ahead markets: Computation of Nash equilibria. Journal of Industrial and Management Optimization, 2015, 11 (3) : 985-998. doi: 10.3934/jimo.2015.11.985 [8] Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control and Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010 [9] Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153 [10] Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51 [11] Sheri M. Markose. Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations. Journal of Dynamics and Games, 2017, 4 (3) : 255-284. doi: 10.3934/jdg.2017015 [12] Janos Kollar. The Nash conjecture for threefolds. Electronic Research Announcements, 1998, 4: 63-73. [13] William Geller, Bruce Kitchens, Michał Misiurewicz. Microdynamics for Nash maps. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1007-1024. doi: 10.3934/dcds.2010.27.1007 [14] Elvio Accinelli, Bruno Bazzano, Franco Robledo, Pablo Romero. Nash Equilibrium in evolutionary competitive models of firms and workers under external regulation. Journal of Dynamics and Games, 2015, 2 (1) : 1-32. doi: 10.3934/jdg.2015.2.1 [15] Shunfu Jin, Haixing Wu, Wuyi Yue, Yutaka Takahashi. Performance evaluation and Nash equilibrium of a cloud architecture with a sleeping mechanism and an enrollment service. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2407-2424. doi: 10.3934/jimo.2019060 [16] Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 [17] Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123 [18] Filipe Martins, Alberto A. Pinto, Jorge Passamani Zubelli. Nash and social welfare impact in an international trade model. Journal of Dynamics and Games, 2017, 4 (2) : 149-173. doi: 10.3934/jdg.2017009 [19] Fernando Luque-Vásquez, J. Adolfo Minjárez-Sosa. Average optimal strategies for zero-sum Markov games with poorly known payoff function on one side. Journal of Dynamics and Games, 2014, 1 (1) : 105-119. doi: 10.3934/jdg.2014.1.105 [20] Xiaolin Xu, Xiaoqiang Cai. Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium. Journal of Industrial and Management Optimization, 2008, 4 (4) : 843-859. doi: 10.3934/jimo.2008.4.843

Impact Factor: