Research update: Gately values

Recently, I stumbled into research on a solution concept for cooperative games that was completely unknown to me: The Gately Value. This solution concept was introduced by Gately (1974) for a particular limited set of cost problems. These cost problems can be transcribed as cooperative games and Gately proposed an innovative solution method to allocate costs and benefits over the participating parties. Gately’s aim was to identify a Core allocation that would satisfy additional fairness properties and that was less complex than Schmeidler’s (1969) nucleolus.

Gately introduced the notion of an individual player’s propensity to disrupt, expressing the relative disruption an individual player causes when leaving the negotiations concerning the allocation of costs and benefits among the players in the game. In fact, Gately formulated this “propensity to disrupt” as the ratio of the other players’ collective loss and the individual player’s loss due to disruption of the negotiations. The prevailing solution method aims to minimise the maximal propensity to disrupt over all allocations and players in the game.

Unfortunately, Gately only considered cost games with three players. Gately’s aims were limited to this class of cooperative games. Only very recently, Staudacher and Anwander (2019) picked up where Gately left off. They focussed mainly on the question whether Gately’s construction method has a solution at all. They showed that the construction method based on Gately’s propensity to disrupt indeed results in a unique solution, thus defining a proper value on the class of regular cooperative games.

Gately’s solution concept falls within the category of a “bargaining-based” solution concepts that also encompasses, e.g., the bargaining set, the Kernel, and the Nucleolus. Contrary to many of these bargaining-based solution concepts, Gately’s conception results in an easily to compute allocation rule that can also be categorised as a compromise value such as the CIS-value and the 𝜏 -value. These solution concepts have a fundamentally different axiomatic foundation than the fairness-based allocation rules such as the egalitarian solution, the Shapley value, the Banzhaf value, and related notions.

Together with Lina Mallozzi, I have further investigated the Gately value for the class of regular cooperative games. We investigated in particular the relationship between the Core and the Gately value as well as under which conditions the Gately value coincides with the Shapley value, which is the prevailing, prime solution concept in cooperative game theory.

This research is reported in two closely related papers:

• Gately Values of Cooperative Games – This paper investigates the Gately value for arbitrary cooperative games, leaving the restriction of three player games. We also consider a generalisation of the Gately value based on a parameter-based quantification of the propensity to disrupt. Foremost, we investigate the relationship of these generalised Gately values with the Core and the Nucleolus and show that Gately’s solution is indeed in the Core for all regular 3-player games. We identify exact conditions under which generally these Gately values are Core imputations for arbitrary regular cooperative games. Finally, we investigate the relationship of the Gately value with the Shapley value.
• Game theoretic foundations of the Gately power measure for directed networks – This paper considers the application of the Gately value to the measurement of power in directed or “hierarchical” networks as descriptors of authority and control relationships among individuals. We introduce a new network centrality measure founded on the Gately value for cooperative games with transferable utilities. The power distribution of a hierarchical network can be represented through a TU-game. We investigate the properties of this TU-representation and investigate the Gately value of the TU-representation resulting in the Gately power measure. We establish when the Gately measure is a Core power gauge, investigate the relationship of the Gately with the 𝛽-measure, and construct an axiomatisation of the new Gately measure.

The Gately value is an interesting concept that interprets a cooperative game in a rather different way than traditional ways such as the Core and the Shapley value. Value generation is approached more as a collective or cooperative process than these traditional concepts. This is most visible in the application of the Gately value to network power measurement as pursued in our second paper. The control of individuals in a hierarchical network is viewed as a collective resource that is divided proportionally among the controlling individuals. This is fundamentally different from the (standard) measurement founded on the Shapley value in which control is completely individualised.

Clearly further research is necessary to flesh out the benefits of the collective approach embodied in solution concepts such as the Gately value.

I refer to the linked working papers posted in my research pages here for more details and examples.