Shapley Shubik Research Articles

Cross invariance, the Shapley value, and the Shapley–Shubik power index

In this paper we propose a simple axiom which, along with the axioms of additivity (transfer) and dummy player, characterizes the Shapley value (the Shapley–Shubik power index) on the domain of TU (simple) games. The new axiom, cross invariance, demands payoff invariance on symmetric players across “quasi-symmetric games,” that is, games where excluding null players, all players are symmetric. Additionally, we demonstrate that the axiom of additivity can be replaced by a new axiom called strong monotonicity, or it can be completely dropped if a stronger version of cross invariance is employed. We also show that the weighted Shapley values can be characterized using a weighted variant of cross invariance. Efficiency is derived rather than assumed in our characterizations. This fresh perspective contributes to a deeper understanding of the Shapley value and its applicability.

Influence relation in two-output multichoice voting games

The influence relation, defined within the set of simple games, is identified as a preorder. Additionally, it is proved to be a subpreorder of the preorders induced by the Shapley–Shubik and Banzhaf–Coleman indices. When this relation extends to voting games with abstention, detailed in Tchantcho et al. (2008), and further to multichoice voting games as in Pongou et al. (2014), it is shown that these extensions aren't always preorders. Even when they are, they don't necessarily align with the preorders induced by the extended Banzhaf–Coleman and Shapley–Shubik power indices in Freixas (2005a) and Freixas (2005b). In this paper, we introduce extensions for two-output multichoice voting games that are both preorders and subpreorders of the Banzhaf–Coleman power index defined in Freixas (2005b). Further, we characterize the two-output multichoice voting games for which one of these new power theories agrees with the generalized Banzhaf–Coleman and Shapley–Shubik power indices in Freixas (2005a) and Freixas (2005b) respectively.

Comparative advantage with many goods: New treatment and results

This paper constitutes the very first treatment of the Shapley–Shubik (1977) market-game mechanism with a continuum of commodities. We develop an oligopolistic-competition model in which product prices are endogenously determined, via buyers’ and sellers’ strategic decisions, and we lay down and examine its mathematical structure. Taking agents’ market power into account, we restudy the Ricardian Law of Comparative Advantage in a many-commodity framework, and obtain a (new) result that is in line with what is perceived in real-world markets: when agents act strategically, they do not specialise based on comparative advantages. For a large class of utility functions, we prove the existence of equilibria at all of which trade is driven neither by absolute nor comparative advantages, but exclusively by strategic decision-making.

Logic, Game Theory, and Social Choice: What Do They Have in Common?

The answer to the question above is that in all these domains axiomatic characterizations are given of, respectively, mathematical reasoning, certain notions from game theory, and certain social choice rules. The meaning of the completeness theorem in logic is that mathematical reasoning can be characterized by a handful of certain (logical) axioms and rules. If we apply mathematical reasoning to elementary arithmetic, i.e., the addition and multiplication of natural numbers, it turns out that almost all true arithmetical statements, for instance, ∀x∀y[x+y=y+x], can be logically deduced from the axioms of Peano. However, in 1931 Kurt Gődel showed that the axioms of Peano do not (fully) characterize the addition and multiplication of the natural numbers, more precisely, that there are certain special self-referential arithmetical sentences that, although true, cannot be deduced from Peano’s axioms. There are axiomatic characterizations of several social choice and ranking rules that say that a given rule is the only one satisfying a particular set of axioms. Arrow’s impossibility theorem in social choice theory tells us that a certain set of, at first sight, quite reasonable axioms for a social ranking rule turns out to be inconsistent. Consequently, a social ranking rule that satisfies the axioms in question cannot exist. Finally, many notions from game theory, such as the Shapley–Shubik and the Banzhaf index, may also be characterized by a set of axioms.

The blocker postulates for measures of voting power

A proposed measure of voting power should satisfy two conditions to be plausible: first, it must be conceptually justified, capturing the intuitive meaning of what voting power is; second, it must satisfy reasonable postulates. This paper studies a set of postulates, appropriate for a priori voting power, concerning blockers (or vetoers) in a binary voting game. We specify and motivate five such postulates, namely, two subadditivity blocker postulates, two minimum-power blocker postulates, each in weak and strong versions, and the added-blocker postulate. We then test whether three measures of voting power, namely the classic Penrose–Banzhaf measure, the classic Shapley–Shubik index, and the newly proposed recursive measure, satisfy these postulates. We find that the first measure fails four of the postulates, the second fails two, while the third alone satisfies all five postulates. This work consequently adds to the plausibility of the recursive measure as a reasonable measure of voting power.

A Matching Model of Co-Residence with a Family Network: Empirical Evidence from China

Abstract We develop a co-residence model between young adults and the elderly as an application of the Shapley–Shubik–Becker bilateral matching framework. This model captures competition between adult children and between parents and parents-in-law. Using microdata from China, we estimate our model by using a network simulation method to fill in partially unobservable marriage links. We find that our model explains the child-side and parent-side competitions observed in the data better than two alternative multinomial logit models with only one-sided competition. In addition, counterfactual experiments quantify the effects of changes in the one-child policy and housing prices on intergenerational co-residence.

Monte Carlo Methods for the Shapley–Shubik Power Index

This paper deals with the problem of calculating the Shapley–Shubik power index in weighted majority games. We propose an efficient Monte Carlo algorithm based on an implicit hierarchical structure of permutations of players. Our algorithm outputs a vector of power indices preserving the monotonicity, with respect to the voting weights. We show that our algorithm reduces the required number of samples, compared with the naive algorithm.

Strategic trading and Ricardian comparative advantage

This paper analyses the failure of the traditional Ricardo–Haberlerian (1817; 1936) theory of comparative advantage (RTCA) in a strategic market game à la Shapley–Shubik (1977). In this model, trade is driven, not by comparative advantages, but by strategic considerations. We prove, in a Ricardian framework, the simultaneous existence of two types of equilibria, at both of which active international trade takes place. In the first type of equilibrium, both countries specialise based on comparative advantages. In the other, eachcountry produces only its comparative-disadvantage good. The welfare properties, and policy implications of this result (using the examples of the China–US trade war and Venezuela), are discussed at length in two dedicated sections. We show that the predictions of the RTCA depend, not on the number of agents in the economy, but on the nature of agents: the RTCA fails to obtain even with an infinite number of large players in each country. We prove that the RTCA prevails when agents are price-takers, and establish the conditions under which equilibria of our market game coincide with Walrasian ones.

Stable coalition structures and power indices for majority voting

AbstractAn (n,k)‐game is a voting game in which each player has exactly one vote, and decisions are made by at least k affirmative votes of the n players. A power index shows the a priori power of the n voters. The purpose of this paper is to show what axioms of power indices generate stable coalition structures for each (n,k)‐game. Using the stability notion of the core, we show that a coalition structure containing a minimal winning coalition is stable for a wide range of general power indices satisfying a set of axioms, such as the Shapley–Shubik, Banzhaf, normalized Banzhaf, and Deegan–Packel power indices. Moreover, we also show that a coalition structure that represents a two‐party system can be stable if the two large parties are close enough in size. Some unstable coalition structures are also analyzed.

Dynamic Programming for Computing Power Indices for Weighted Voting Games with Precoalitions

We study the efficient computation of power indices for weighted voting games with precoalitions amongst subsets of players (reflecting, e.g., ideological proximity) using the paradigm of dynamic programming. Starting from the state-of-the-art algorithms for computing the Banzhaf and Shapley–Shubik indices for weighted voting games, we present a framework for fast algorithms for the three most common power indices with precoalitions, i.e., the Owen index, the Banzhaf–Owen index and the symmetric coalitional Banzhaf index, and point out why our new algorithms are applicable for large numbers of players. We discuss implementations of our algorithms for the three power indices with precoalitions in C++ and review computing times, as well as storage requirements.

Evaluation of decision power in multi-dimensional rules

This work deals with the evaluation of decision power in multi-dimensional rules. Courtin and Laruelle (2020) introduced a decision process that specifies the collective acceptance or rejection of a proposal with several dimensions. The decision process is modeled as follows: (i) There are several individuals. (ii) There are several dimensions. (iii) Each of the individuals expresses a binary choice (“Yes” or “No”) on each dimension. (iv) A decision process maps each choice to a final binary decision (“Yes” or “No”). We extend and characterize six well-known power indices within this context: the Shapley–Shubik index (Shapley and Shubik, 1954), the Banzhaf index (Banzhaf, 1965), the Public good index (Holler, 1982), the Null individual free index (Alonso-Meijide et al., 2011), the Shift index (Alonso-Meijide and Freixas, 2010) and the Deegan–Packel index (Deegan and Packel, 1978).

Brexit and Power in the Council of the European Union

The exit of the United Kingdom from the European Union has had profound economic and political effects. Here, we look at a particular aspect, the power distribution in the Council of the European Union. Using the Shapley–Shubik power index, we calculate the member states’ powers with and without the United Kingdom and update earlier power forecasts using the Eurostat’s latest population projections. There is a remarkably sharp relation between population size and the change in power: Brexit increases the largest members’ powers while decreasing the smallest ones’ powers.

The Public Good Spatial Power Index in political games

We propose a new spatial index called the Public Good spatial index, which is the spatial version of the standard Public Good index under independence behaviors (PGI). However, we show that the spatial model used, is not well suited for both, the standard Shapley Shubik index and the standard Public Good index under Homogeneity behaviors (PGH); and consequently they do not have a consistent spatial version with respect to our model. By contrast, the same spatial model is appropriated for the Banzhaf index and the aforementioned PGI index, concomitantly, allowing their spatial versions in political games. We also argue that those two well convenient spatial indices are only understandable under behavioral descriptions with independence assumptions on political issues. Finally, the paper details our findings by means of examples, comparisons, and it also provides relevant ways of computing spatial power indices in real case studies when it comes to lower dimensions.

Axiomatizations for the Shapley–Shubik power index for games with several levels of approval in the input and output

The Shapley-Shubik index is a specialization of the Shapley value and is widely applied to evaluate the power distribution in committees drawing binary decisions. It was generalized to decisions with more than two levels of approval both in the input and the output. The corresponding games are called $(j,k)$ simple games. Here we present a new axiomatization for the Shapley-Shubik index for $(j,k)$ simple games as well as for a continuous variant, which may be considered as the limit case.

Dynamic programming algorithms for computing power indices in weighted multi-tier games

In weighted games each voter has a weight assigned and “yes”-voters win if the sum of each weight is greater than or equal to the quota. In weighted multi-tier games, we have several weighted games (tiers) over the same set of voters. In this article, algorithms for calculating Banzhaf and Shapley–Shubik indices for three different type of games are proposed: weighted AND-games, weighted OR-games and games where the tiers have conjunctive normal form. The presented algorithms are generalizations of known computational methods using dynamic programming technique. Finally, some applications and experiments are carried out and these algorithms are compared to a fairly new method based on binary decision diagrams.

Allocation of risk capital in a cost cooperative game induced by a modified expected shortfall

The standard theory of coherent risk measures fails to consider individual institutions as part of a system which might itself experience instability and spread new sources of risk to the market participants. This paper fills this gap and proposes a cooperative market game where agents and institutions play the same role. We take into account a multiple institutions framework where some institutions jointly experience distress, and evaluate their individual and collective impact on the remaining institutions in the market. To carry out the analysis, we define a new risk measure (SCoES) which is a generalization of the Expected Shortfall of and we characterize the riskiness profile as the outcome of a cost cooperative game played by institutions in distress. Each institution’s marginal contribution to the spread of riskiness towards the safe institutions in then evaluated by calculating suitable solution concepts of the game such as the Banzhaf–Coleman and the Shapley–Shubik values.

On trade in bilateral oligopolies with altruistic and spiteful agents

This paper studies the effects of altruism and spitefulness in a two-sided market in which agents behave strategically and trade according to the Shapley–Shubik mechanism. By assuming that altruistic agents have concerns for others on the opposite side of the market, it shows that agents always find advantageous to trade. However, they prefer to stay out of the market and consume their endowments when there are altruistic agents who have concerns for the welfare of those on the same side of the market, or when there are spiteful agents. These non-trade situations occur either because the necessary first-order conditions for optimality are violated or because agents’ payoff functions are not concave.

The vertex cover game: Application to transport networks

As communications are progressing, transport networks need to be monitored, more specifically, camera surveillance of violations is needed. Standards are being developed on how to install cameras, and the question of efficiently distributing surveillance devices across the road network ensue. This task is addressed in this paper by using the methods of the cooperative game theory. The vertex cover game is introduced, and its properties are studied. Since surveillance cameras are to cover all areas of the network, the characteristic function depends on the vertex covers of the graph. The Shapley–Shubik index is used as the measure of centrality. The Shapley–Shubik index is shown to be efficient in a vertex cover game for the allocation of cameras in a transport network. Proceeding from the Shapley–Shubik indices calculated in this study, recommendations were given for the allocation of surveillance cameras in a specific transport network in a district of the City of Petrozavodsk, Russia.

The Coleman–Shapley index: being decisive within the coalition of the interested

The Coleman power of a collectivity to act (CPCA) is a popular statistic that reflects the ability of a committee to pass a proposal. Applying the Shapley value to that measure, we derive a new power index—the Coleman–Shapley index (CSI)—indicating each voter’s contribution to the CPCA. The CSI is characterized by four axioms: anonymity, the null voter property, the transfer property, and a property stipulating that the sum of the voters’ power equals the CPCA. Similar to the Shapley–Shubik index (SSI) and the Penrose–Banzhaf index (PBI), our new index reflects the expectation of being a pivotal voter. Here, the coalitional formation model underlying the CPCA and the PBI is combined with the ordering approach underlying the SSI. In contrast to the SSI, voters are ordered not according to their agreement with a potential bill, but according to their vested interest in it. Among the most interested voters, power is then measured in a way similar to the PBI. Although we advocate the CSI over the PBI so as to capture a voter’s influence on whether a proposal passes, our index gives new meaning to the PBI. The CSI is a decomposer of the PBI, splitting the PBI into a voter’s power as such and the voter’s impact on the power of the other voters by threatening to block any proposal. We apply our index to the EU Council and the UN Security Council.

An approach via generating functions to compute power indices of multiple weighted voting games with incompatible players

We introduce a new generating function based method to compute the Banzhaf, Deegan–Packel, Public Good (a.k.a. the Holler power index) and Shapley–Shubik power indices in the presence of incompatibility among players. More precisely, given a graph $$G=\left( V,E\right) $$ with V the set of players and E the edge set, our extension involves multiple weighted voting games (MWVG’s) and incompatible players, i.e., pairs of players belonging to E are not allowed to cooperate. The route to obtain the aforementioned generating functions comprises the use of a key lemma characterizing the set of minimal winning coalitions of the game with incompatibility due to Alonso-Meijide et al. (Appl Math Comput 252(1):377–387, 2015), a tool from combinatorial analysis, namely, the Omega calculus in partition analysis, and basic tools borrowed from commutative algebra involving the computation of certain quotients of polynomial rings module polynomial ideals. Using partition analysis, we obtain new generating functions to compute the Deegan–Packel and Public Good power indices with incompatibility leading to lower time complexity than previous results of Chessa (TOP 22(2):658–673, 2014) and some results of Alonso-Meijide et al. (Appl Math Comput 219(8):3395–3402, 2012). Using a conjunction of partition analysis and commutative algebra, we extend to MWVG’s the generating function approach to compute the Banzhaf and Shapley–Shubik power indices in the presence of incompatibility. Finally, an example taken from the real-world, i.e., the European Union under the Lisbon Treaty, is used to illustrate the usefulness of the Omega package, a symbolic computational package that implements the Omega calculus in Mathematica, due to Andrews et al. (Eur J Comb 22(7):887–904, 2001) in the context of MWVG’s by computing the PG power index of the associated voting game.