A Sequential Allocation Problem: The Asymptotic Distribution of Resources

Our problem can be viewed as a particular case of a more general river sharing problem. Since the axioms and definitions that we use in our framework may vary from those of a more general river sharing problem, it is treated as an independent problem of its own. There are also similarities with the well-known dictator game of Kahneman et al. (1986), the ultimatum game of Güth et al. (1982) or the airport problem ( Littlechild and Owen 1973), see below.

Ambec et al. (2013) address the vulnerability and monitoring difficulties associated with the compliance of existing water sharing arrangements.

Carraro et al. (2007) and Ambec and Ehlers (2008) survey the literature on non-cooperative and cooperative solutions for the river sharing problem. Parrachino et al. (2006) review the literature.

See Ambec and Ehlers (2008) for an extension of downstream incremental distribution for single-peaked preferences. See also Kilgour and Dinar (2001) and Wang (2011), among others.

Utility assumptions are subjective with profound implications on the results. Dinar et al. (1992) critique the use of game theoretical based transfers that are not related to market prices, and the representation of the problem in the “utility-space”. More recently, the same position is defended in Houba (2008).

Nonetheless, we will show a connection between the allocation rule proposed in the present paper and the Shapley (1953) value of a particular TU game with a structure similar to an airport type problem (Littlechild and Owen 1973; Thomson et al. 2007).

Axioms 1 and 2 impose the following payoff bounds,The set of admissible payoff profiles that satisfy these bounds is uncountable. This aspect leads to some technical issues that are addressed later. Table 1 presents the (unit) discretized set of admissible payoff profiles for \(n=3\) and \(E=3,6,9,12.\)

The reader is free to consider other interpretations. There is no indisputable definition of “representativeness” and “equal treatment”. “Indisputable”, is also a not well-defined concept.

Note the analogy between equal weighting and the Shapley (1953) value in which each individual marginal contributions to the coalition is equally weighted. The difference with respect to the Shapley value is that we do not think in terms of forming meaningful coalitions but in terms of admissible allocations. In Sect. 6 below we formalize the connection between these two concepts.

The need of a discrete action space is also motivated by the problem of defining arbitrarily small or large values on a real number system.

Following the discussion, with a discrete action space, we can rewrite the bounds in footnote 8 as \(c_{1}\in \left[ m,nm-1\right] ,\ \)and \( c_{i}\in \left[ 1,nm/i\right] ,\) for \(i\in N\backslash 1.\)

Properties P1,  P2 and P3 receive empirically support in Bahr and Requate (2014), Bonein et al. (2007) and Engel (2011). Engel (2011) shows that if exist more than one recipient the dictator naturally accepts a lower allocation for himself.

The equity concerns of the other individuals toward individual \(i=1\) is negative and equals to \(-(1\,-\,\phi _{1}^{n}),\) while the strategic positioning value is maximal and equal to 1. On the other hand, the strategic positioning value of individual n is minimal and equals to 0,  but benefits from positive equity concerns, which has a value of \(\phi _{n}^{n}.\) The distinction between equity concerns and strategy positioning value should be object of further research.

We are thankful to an anonymous referee for pointing out this property and the description of the associated TU game.

In spite that the Shapley value and our approach agree for the described game, there is no immediate link between the marginal contribution of each different sized coalition with each term in the summation \( \sum \nolimits _{k=i}^{n}1/k\) of expression (2) in a general and meaningful way.

In the case \(n=2\) the asymptotic distribution remains the same as in 3. For isntance, if Axiom 1 is replaced by Axiom 5, we simply remove the payoff profile \(\left( m,m\right) \) from the admissible set, which appears only once for any m. If Axiom 2 is replaced by Axiom 6, we add the payoff profile \(\left( 2m,0\right) .\) Asymptotically, a single term is irrelevant. However, for \( n\ge 3\) we must have different asymptotic distributions, because the removed and/or added allocation profiles increase with m.

Another possible extension is to verify the relation between the asymptotic approach of the current paper with physics concepts such as the centroid, center of mass or gravity. Note that each admissible allocation is uniformly weighted (Axiom 4) but the distribution of mass is not homogeneous because the allocations are heterogeneous. Therefore, it is likely that our result is equivalent to the center of mass of the admissible set. Such a result requires knowledge about the mass density function. However, care should be taken, the relaxation of an axiom (a simple inequality) leads to a different admissible set, which is may not be captured by an integration-based method. We are thankful to an anonymous referee who pointed out the possibility of this property.

The sequences in this paper have been found by the authors and are registered in the “OEIS Foundation Inc. (2011), The On-Line Encyclopedia of Integer Sequences, http://​oeis.​org”.