Abstract
We consider non-zero sum bi-matrix games where one player presumes the role of a leader in the Stackelberg model, while the other player is her follower. We show that the leader can improve her reward if she can incentivise her follower by paying some of her own utility to the follower for assigning a particular strategy profile. Besides assuming that the follower is rational in that he tries to maximise his own payoff, we assume that he is also friendly towards his leader in that he chooses, ex aequo, the strategy suggested by her—at least as long as it does not affect his expected payoff. Assuming this friendliness is, however, disputable: one could also assume that, ex aequo, the follower acts adversarially towards his leader. We discuss these different follower behavioural models and their implications. We argue that the friendliness leads to an obligation for the leader to choose, ex aequo, an assignment that provides the highest follower return, resulting in ‘friendly incentive equilibria’. For the antagonistic assumption, the stability requirements for a strategy profile should be strengthened, comparable to the secure Nash equilibria. In general, no optimal incentive equilibrium for this condition exists, and therefore we introduce ε-optimal incentive equilibria for this case. We show that the construction of all of these incentive equilibria (and all the related leader equilibria) is tractable.
Highlights
Stackelberg models [1] have been studied in detail in Oligopoly Theory [2]
The players’ utility and social welfare is seen as counterintuitive. e.g., social welfare may arbitrarily come worse and they focus completely on pure strategies, whereas our solution approach tries to increase the payoff of both players, which often leads to a good social outcome and the equilibrium we study is mixed only for the leader
We summarise the results for friendly incentive equilibria (IE) and leader equilibria (LE), respectively, in Table for continuous variables in the 0 to 1 range, and in Table for integer variables in the −10 to 10 range
Summary
Stackelberg models [1] have been studied in detail in Oligopoly Theory [2]. In a Stackelberg model, one player or firm acts as a market leader and the other is a follower. The strategy profile ( I I, I I ) is the leader equilibrium that is not a Nash equilibrium It provides a payoff of 2 and 1 to the leader and the follower, respectively. Nash equilibrium here is the strategy profile ( D, D ) with a joint return of −16 Another observation is that leader equilibria are not powerful enough to overcome this antinomy. Co-operating becomes an optimal choice for her follower In this example, (C, C ) with bribery value 1 is the only incentive equilibrium. The leader has to select strategy profiles that are secure, they might not be optimal. (the relative loss of the leader is arbitrarily small, such that we skip over this detail in the remainder of the introductory part.)
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