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. 
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C. D. AliprantisD. 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.

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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.

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D. Glycopantis, On games with imperfect recall and games with perfect recall, SSRN-id2520455, (2014).

[8] P. Morris, Introduction to Game Theory, Springer, 1994. 
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[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. AliprantisD. 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. 
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