Rewarding moderate behavior in a dynamic Nash demand game
The paper studies a repeated Nash demand game (rNDG) whose particular version is widely known as the divide-the-dollar game. Let \(S\subseteq\mathbb{R}^2\) be a feasible set of utility allocations among two players. Assume that: (o) \(S\subseteq [0,\infty)\times [0,\infty)\); (i) \(S\) is compact; (ii) \(S\) is convex; (iii) weak and strong Pareto frontiers of \(S\) coincide; thus, every allocation \((s_1,s_2)\in S\) which cannot be improved for both players, cannot be also improved for one player without harming the other; (iv) \(S\) is comprehensive, i.e., \([0,s_1]\times [0,s_2] \subseteq S\) when \((s_1,s_2)\in S\); (v) \(S\) is regular (egalitarian point is utilitarian), i.e., \((e(S),e(S))\in \arg\max_{(s_1,s_2)\in S}\;(s_1+s_2)\) for \(e(S) = \max \{x: (x,x)\in S\}\). The rNDG is played as follows: (I) at every stage \(t\) both players declare their demands \((x_t,y_t)\); (II) if \((x_t,y_t)\in S\), then the players receive \(x_t\) and \(y_t\), respectively, and the game stops; (III) else \(m=\min(x_t,y_t)\) is offered to a less greedy player (chosen by a toss of a symmetric coin in case of the tie \(x_t=y_t\)); the other player has two options; (IV) either the other player accepts the rest \(f(m)\) which in conjunction with \(m\) is Pareto efficient in \(S\), and the game stops; (V) or the other player discards the offered rest \(f(m)\), and so the game enters next stage \(t+1\). An egalitarian profile of \(S\) with discounting \(\delta_1, \delta_2 \in(0,1)\) is built on the following behaviour: (1) a player demands \(e(S)\) at each stage; (2) when found in a greedy position, player \(i\) (\(i=1,2\)), who demanded \(s_i=e(S)\), accepts the offered Pareto optimal rest \(f(s_{3-i})\) from the demand \(s_{3-i}\leq e(S)\) of the other player, provided \(f(s_{3-i}) \geq \delta_i \cdot e(S)\). The main result of the paper states that the rNDG (I)--(V) with discounting \(\delta_1, \delta_2\in (0,1)\), played to satisfy demands according to a feasible set \(S\) which fulfills properties (o)--(v), admits a unique subgame perfect equilibrium which turns out to be an egalitarian profile. Moreover, if \(S\) does not satisfy regularity (v), then the rNDG on \(S\) still admits a unique equilibrium, but this time an equilibrium is required to satisfy monotone acceptance: the offers which are higher than good offers in the past are considered good any time later. Finally, an example of the rNDG on a non-convex \(S\) is considered so that there exists an equilibrium which fails to obey monotone acceptance. The problem of whether such a phenomenon occurs for a convex non-regular \(S\) appears to be open.
scientific article; zbMATH DE number 7219726
scientific article