As is well known, zero-sum games are appropriate instruments for the analysis of several issues across areas including economics, international relations and engineering, among others. In particular, the Nash equilibria of any two-player finite zero-sum game in mixed-strategies can be found solving a proper linear programming problem. This chapter investigates and solves non-Archimedean zero-sum games, i.e., games satisfying the zero-sum property allowing the payoffs to be infinite, finite and infinitesimal. Since any zero-sum game is coupled with a linear programming problem, the search for Nash equilibria of non-Archimedean games requires the optimization of a non-Archimedean linear programming problem whose peculiarity is to have the constraints matrix populated by both infinite and infinitesimal numbers. This fact leads to the implementation of a novel non-Archimedean version of the Simplex algorithm called Gross-Matrix-Simplex. Four numerical experiments served as test cases to verify the effectiveness and correctness of the new algorithm. Moreover, these studies helped in stressing the difference between numerical and symbolic calculations: indeed, the solution output by the Gross-Matrix Simplex is just an approximation of the true Nash equilibrium, but it still satisfies some properties which resemble the idea of a non-Archimedean ε-Nash equilibrium. On the contrary, symbolic tools seem to be able to compute the “exact” solution, a fact which happens only on very simple benchmarks and at the price of its intelligibility. In the general case, nevertheless, they stuck as soon as the problem becomes a little more challenging, ending up to be of little help in practice, such as in real time computations. Some possible applications related to such non-Archimedean zero-sum games are also discussed.

Cococcioni, M., Fiaschi, L., Lambertini, L. (2022). Computing Optimal Decision Strategies Using the Infinity Computer: The Case of Non-Archimedean Zero-Sum Games. Heidelberg : Springer [10.1007/978-3-030-93642-6_11].

### Computing Optimal Decision Strategies Using the Infinity Computer: The Case of Non-Archimedean Zero-Sum Games

#### Abstract

As is well known, zero-sum games are appropriate instruments for the analysis of several issues across areas including economics, international relations and engineering, among others. In particular, the Nash equilibria of any two-player finite zero-sum game in mixed-strategies can be found solving a proper linear programming problem. This chapter investigates and solves non-Archimedean zero-sum games, i.e., games satisfying the zero-sum property allowing the payoffs to be infinite, finite and infinitesimal. Since any zero-sum game is coupled with a linear programming problem, the search for Nash equilibria of non-Archimedean games requires the optimization of a non-Archimedean linear programming problem whose peculiarity is to have the constraints matrix populated by both infinite and infinitesimal numbers. This fact leads to the implementation of a novel non-Archimedean version of the Simplex algorithm called Gross-Matrix-Simplex. Four numerical experiments served as test cases to verify the effectiveness and correctness of the new algorithm. Moreover, these studies helped in stressing the difference between numerical and symbolic calculations: indeed, the solution output by the Gross-Matrix Simplex is just an approximation of the true Nash equilibrium, but it still satisfies some properties which resemble the idea of a non-Archimedean ε-Nash equilibrium. On the contrary, symbolic tools seem to be able to compute the “exact” solution, a fact which happens only on very simple benchmarks and at the price of its intelligibility. In the general case, nevertheless, they stuck as soon as the problem becomes a little more challenging, ending up to be of little help in practice, such as in real time computations. Some possible applications related to such non-Archimedean zero-sum games are also discussed.
##### Scheda breve Scheda completa Scheda completa (DC)
2022
Numerical Infinities and Infinitesimals in Optimization
271
295
Cococcioni, M., Fiaschi, L., Lambertini, L. (2022). Computing Optimal Decision Strategies Using the Infinity Computer: The Case of Non-Archimedean Zero-Sum Games. Heidelberg : Springer [10.1007/978-3-030-93642-6_11].
Cococcioni, Marco; Fiaschi, Lorenzo; Lambertini, Luca
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/915001`
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