Ines Lindner

L.S. Penrose's limit theorem: proof of some special cases

L.S. Penrose was the first to propose a measure of voting power (which later came to be known as ‘the [absolute] Banzhaf (Bz) index’). His limit theorem—which is implicit in his booklet (1952) and for which he gave no rigorous proof—says that in simple weighted voting games (WVGs), if the number of voters increases indefinitely while the quota is pegged at half the total weight, then—under certain conditions—the ratio between the voting powers (as measured by him) of any two voters converges to the ratio between their weights. We conjecture that the theorem holds, under rather general conditions, for large classes of variously defined WVGs, other values of the quota, and other measures of voting power. We provide proofs for some special cases.

Cases where the Penrose Limit Theorem does not hold

Abstract Penroses limit theorem (PLT, really a conjecture) states that the relative power measure of two voters tends asymptotically to their relative voting weight (number of votes). This property approximately holds in most of real life and in randomly generated WVGs for various measures of voting power. Lindner and Machover prove it for some special cases; amongst others they give a condition for this theorem to hold for the Banzhaf–Coleman index for a quota of 50%. We show here, by counterexamples, that the conclusion need not hold for other values of the quota. In doing this, we present an analytic proof of a counterexample recently given by Chang et al. using simulation techniques.

Social Fractionalization, Endogenous Appropriation Norms, and Economic Development

We investigate how social composition affects competitive and cooperative behaviour in a linear growth model without secure property rights. If a society is homogeneous or highly fractionalized, it is in the self-interest of people to cooperate. The first-best allocation is enforced through trigger strategies, and growth is independent from social structure. If a society is polarized, i.e. if it consists of a small number of groups, the first-best solution can turn out to be unenforceable and groups will follow an exploitative strategy. In this case, the rate of growth is monotonously decreasing in the degree of fractionalization. Copyright (c) The London School of Economics and Political Science 2007.

Networks and Collective Action

This paper proposes a new measure for a groups ability to lead society to adopt their standard of behavior, which in particular takes account of the time the group takes to convince the whole society to adopt their position. This notion of a groups power to initiate action is computed as the reciprocal of the resistance against it, which is in turn given by the expected absorption time of a related finite state partial Markov chain that captures the social dynamics. The measure is applicable and meaningful in a variety of models where interaction between agents is formalized through (weighted) binary relations. Using Percolation Theory, it is shown that the group power is monotonic as a function of groups of agents. We also explain the differences between our measure and those discussed in the literature on Graph Theory, and illustrate all these concerns by a thorough analysis of two particular cases: the Wolfe Primate Data and the 11S hijackers’ network.

The Power of a Collectivity to Act in Weighted Voting Games with Many Small Voters

We analyze the propensity to approve a random proposal of a large committee that makes decisions by weighted voting. The approach is a generalized version of James Coleman’s “power of a collectivity to act”. Throughout the paper it is assumed that the voters are of two kinds: a fixed (possibly empty) set of “major” (big) voters with fixed weights, and an ever-increasing number of “minor” (small) voters, whose total weight is also fixed, but where each individual’s weight becomes negligible. As our main result, we obtain that asymptotically many minor voters act like a modification of the quota for the vote among major voters. The paper estimates the rate of convergence which turns out to be very high if the weight distribution among the small voters is not too skewed. The results obtained are illustrated by evaluating the decision rules for the Council of Ministers of the EU for various scenarios of EU enlargement.

Modified Power Indices for Indirect Voting

The Electoral College remains a controversial feature of U.S. political decision-making. After most U.S. presidential elections, there are calls for passage of a constitutional amendment to either abolish it or to ‘reform’ it substantially. There are numerous complaints about the Electoral College, of which the most important is the potential for the winner of the Electoral College majority to be a popular vote loser. Consider three assertions that often surface in the debates about the political impact of the Electoral College.

Mass-mobilization with noisy conditional beliefs

We examine the role played by information in shaping popular expectations in the process of political mobilization, and the development of revolutionary movements in particular. The vast majority of the people face the trade-off between the ‘risky’ option of joining the revolutionary movement and the ‘safe’ option of supporting the present regime. The large majority of the people are influenced by the expectations of a successful campaign of either political side. Only if the expected or perceived support for the opposition exceeds a critical mass, does the revolution become effective. We show that the growth-dynamics of revolutionary movements heavily depends on the parameters determining the perception of the actual state of support. And, we will argue that it is the informational aspect that provides the necessary bias to get out of the self-confirming outsider position. Especially, as we argue, it provides an explanation for the symbolic violence used by terrorist groups as it serves them as an instrument by which the mobilization dynamics is turned into a format where information has the largest impact. Surprisingly, this will in general be the situation where a majority of the people dissents from the acts of terror.

A simple “market value” bargaining model for weighted voting games: characterization and limit theorems

We offer a bargaining model for weighted voting games that is a close relative of the nucleolus and the kernel. We look for a set of weights that preserves winning coalitions that has the property of minimizing the difference between the weight of the smallest and the weight of the largest Minimum Winning Coalition. We claim that such a set of weights provides an a priori measure of a weighted voter’s bribeworthiness or market value. After introducing our model, we provide a characterization result for this model and show its links to other bargaining model approaches in the literature. Then we offer some limit results showing that, with certain reasonable conditions on the distributions of weights, as the size of the voting body increases, the values of bribeworthiness we calculate will approach both the weights themselves and the Banzhaf scores for the weighted voting game. We also show that, even for relatively small groups using weighted voting, such as the membership of the European Council of Ministers (and its predecessors) 1958–2003, similarities among the usual a priori power scores, bribeworthiness/market value, and the weights themselves, will be quite strong.

Innovation and Inequality in a Small World

We present a multi-country theory of economic growth and R&D driven technological progress in which countries are connected by a network of knowledge exchange. Technological progress in any country depends on the state of technology in the countries it exchanges knowledge with. The diffusion of knowledge throughout the world explains a period of increasing world inequality after the take-off of the forerunners of the industrial revolution, followed by decreasing relative inequality. Knowledge diffusion through a Small World network produces an extraordinary diversity of country growth performances, including the overtaking of individual countries and the replacement of the technologically leading country in the course of world development.

Diffusion of behavior in network games with threshold dynamics

In this paper we propose a generalized model of network games to incorporate preferences as an endogenous driving force of innovation. Individuals can choose between two actions: either to adopt a new behavior or stay with the default one. A key element is an individual threshold, i.e. the number or proportion of others who must take action before a given actor does so. This threshold represents an individual’s inclination to adopt the new behavior. The main novelty of the paper is to assume that the thresholds are endogenously determined. Agents change their inclination by exposition to other inclinations in the social network. This provides a coupled dynamical system of aggregate adoption rate and inclinations orchestrated by the network. With our model we are able to explain a variety of adoption behavior. Of particular interest is the existence of non-monotonic behavior of the aggregate adoption rate which is not possible in the benchmark model without inclination. Our model is therefore able to explain “sudden” outbreaks of collective action. This suggests to reinvent the common static and exogenous concept of a tipping point by defining it endogenously generated by the network.