Abstract
In the Nash demand game (NDG) n players announce utility demands, the demands are implemented if they are jointly feasible, and otherwise no one gets anything. If the utilities set is the simplex, the game is called “divide-the-dollar”. Brams and Taylor (Theory Decis 37:211–231, 1994) studied variants of divide-the-dollar, on which they imposed reasonableness conditions. I explore the implications of these conditions on general NDGs. In any reasonable NDG, the egalitarian demand profile cannot be obtained via iterated elimination of weakly dominated strategies. Further, a reasonable NDG may fail to have a Nash equilibrium, even in mixed strategies. In the 2-person case, existence of pure strategy equilibrium is equivalent to the existence of a value, in the sense that each player can secure the egalitarian payoff level independent of his opponent’s play. This result does not extend to reasonable NDGs with more than two players. Interestingly, there are results for reasonable NDGs that hold for two and three players, but not for \(n\ge 4\) players.
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Notes
The papers that will be mentioned in the next paragraph follow the continuous bids approach.
That is, the players face a set of feasible utility allocations, S, they simultaneously announce utility-demands, obtain what they asked for if their announcements are jointly feasible, and nothing otherwise. Nash considered \(n=2\), but the extension to a general n is straightforward.
I refer to Nash’s original game as the classical NDG.
In the abovementioned papers, a player’s own demand is central in determining his payoff. An alternative approach is to have one’s payoff be affected by others’ reports. This peer evaluation approach has been explored by Boudreau and Knoblauch (2011), De Clippel et al. (2008), Knoblauch (2009) and Tideman and Plassmann (2008).
An obvious example is the result mentioned above about the minmax value in zero-sum games, but there are many others; not to drift away from my main thread here, I will not go into a more detailed account. It is worth mentioning, however, that Karagözoğlu and Rachmilevitch (2018) is another “exception to the rule.” There, it is shown that in some DD games, the egalitarian outcome is sustainable in equilibrium if and only if the number of players is at most four.
This is the only place in the paper where I allow for mixed strategies. In the rest of the paper, I focus on pure strategies, which is the common approach to DD/NDG games. A notable exception is Malueg (2010), who characterized mixed strategy equilibria in the classical NDG. Another relevant paper is by Connell and Rasmusen (2019), who characterized mixed strategy equilibria in a DD-like game, in which demands incompatibility leads to null payoffs (like in DD), but in which feasible demands are mapped into the corresponding proportional payoffs (unlike DD). For example, the demands (0.3, 0.4) are not mapped into the payoffs (0.3, 0.4), but into \((\frac{3}{7},\frac{4}{7})\).
Vector inequalities: \(\mathbf{x }R \mathbf{y }\) iff \(\mathbf{x }_i R \mathbf{y }_i\) for all i, for both \(R\in \{<,\le \}\).
See, e.g., Rachmilevitch (2017).
Dufwenberg and Stegeman (2002).
In fact, it is a strongly dominant strategy.
Namely, their model is identical to the one described in Sect. 2 under the assumption that S is the unit simplex, and with the restriction that demand-announcements must lie in some finite set (to be precise, because bids are restricted to a finite set, S is a discrete subset of the unit simplex).
In Rachmilevitch (2017) I showed that in the continuous model, the egalitarian demand need not be dominated.
Namely, at every step in the process all dominated-at-that-step strategies are deleted.
If the egalitarian demand is deleted at the first step then clearly it cannot be obtained at the last step.
For this equilibrium, it is necessary that \(n\ge 3\). As mentioned above, when one starts with the profile \((\frac{1}{n-1},\frac{1}{n-1},\ldots ,\frac{1}{n-1})\), then a unilateral deviation downwards does not change one’s payoff if the new demand is at least \(\frac{1}{n}\)—by (iv); in particular, none of lines (i)–(iii) apply, and the induced payoff vector is identical to the original payoffs, namely \((\frac{1}{n},\ldots ,\frac{1}{n})\). However, for \(n=2\) the profile (1, 1) is not an equilibrium: a deviation downwards to \(x\in (\frac{1}{2},1)\) by, say, player 1, brings about the payoffs \((x,1-x)\). This is because in this case line (i) does apply.
By Proposition 2, equilibrium payoffs are obviously egalitarian when \(n=2\).
Brams and Taylor studied DD1 under the assumption of discrete bids. A detailed account of DD1’s continuous version appears in Rachmilevitch (2017). Here, I settle for an informal description of this game.
Prior to the deviation his payoff is \(\frac{1}{3}-\delta \), after the deviation it is greater than \(\frac{1}{3}\).
I skip formal definition of a mixed strategy, trusting that no confusion will arise.
Note that Proposition 2 cannot be applied here, because mixed strategies are allowed.
When a player takes an action \(a<\frac{1}{2}\) his payoff is a, because the second and fourth lines in the utility’s definition cannot apply. However, an a-payoff contradicts the Claim.
In fact, continuity guarantees the existence of a pure strategy equilibrium.
The abovementioned work of Reny considers Bayesian games, but it applies to DD/NDGs as well; all that one needs to do to apply the existence result is to define the set of each player’s “types” to be a singleton.
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Rachmilevitch, S. Reasonable Nash demand games. Theory Decis 93, 319–330 (2022). https://doi.org/10.1007/s11238-021-09849-6
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DOI: https://doi.org/10.1007/s11238-021-09849-6