Set of Equilibria in Mixed-Strategy for Hierarchical Structures

Abstract

In most socio-economic entities a hierarchical structure can be distinguished. In the process of solving a task, the final result depends on the decisions made at each level. The choice made by a certain actor involved in solving the problem influences the choices of others and, not least, final profit. The paper aims to research mixed-strategy hierarchical games in three-level. That is, the game consists of three players, each of them has two strategies and a gain function. Players make moves in hierarchical mode: first player makes the choice and communicates the result to second player; second player knowing first player's choice, as well as third player's set of strategies and payoff function, makes his move and communicates the outcome to third player; finally, third player knowing the predecessors’ choices, makes his choice. Thus, a situation is created and each player calculates his payoff. It is considered that all players maximize their payoff. The given model includes a wide range of problems that can appear in the socio-economic domain. To computing the Stackelberg equilibria set (SES), reverse induction and the graph reduction of best response mapping of the third player are used. A particular case of the results presented by Lozan and Ungureanu (2010, 2013, 2016, 2018) is studied and concretized. All possible cases for the graph of third player (𝐆𝐫𝟑) are investigated, the construction method is described by Ungureanu and Botnari (2005). Then, for player two, the possibilities that may arise for constructing his graph of best response mapping (𝐆𝐫𝟐) are analyzed. Finally, the first player determines his best moves on 𝐆𝐫𝟐, thus determining the SES in mixed strategies. DOI: https://doi.org/10.53486/cike2023.40; UDC: 330.4:519.86; JEL: C02, C61, C62, C65, C72, C79.