Am empirical comparison of the performance of classical power indices

Power indices are general measures of the relative voting power of individual members of a voting body. They are useful in helping understand and design voting bodies particularly those which employ weighted voting, in which different members having different numbers of votes. It is well known that in such bodies a member's voting power, in the sense of their capacity to affect the outcomes of votes called, rarely corresponds to the actual number of votes allocated to him. Many voting bodies for which this is an important consideration exist: examples include international organisations (notably the World Bank, the IMF, the European Union), the US presidential Electoral College and corporations in which votes are proportionate to stockholdings. Two classical power indices dominate the literature: the Shapley-Shubik index and the Banzhaf index (also known by other names). Both are based on the idea that a member's power depends on the relative number of times they can change a coalition from losing to winning by joining it and adding their vote. They may be defined in probabilistic terms as the probability of being able to swing the result of a vote, where all possible outcomes are taken as equiprobable. The indices differ however in the way they count voting coalitions. In probabilistic terms they use different coalition models and therefore differ in precisely what is meant by equiprobable outcomes. The indices have been used in a number of empirical applications but their relative performance has remained an open question for many years, a factor, which has hindered the wider acceptance of the approach. Where both the indices have been used for the same case, they have often given different results, sometimes substantially so, and theoretical studies of their properties have not been conclusive. There is therefore a need for comparative testing of their relative performance in practical contexts. Very little work of this type has been done however for a number of reasons: lack of independent indicators of power in actual voting bodies with which to compare them, difficulties in obtaining consistent data on a voting body over time with sufficient variation in the disposition of votes among members of actual legislatures and the lack of independent criteria against which the results of the indices may be judged. It has also been hampered to some extent by lack of easily available algorithms for computing the indices in large games. This paper assesses the indices against a set of reasonable criteria in terms of shareholder voting power and the control of the corporation in a large cross section of British companies. Each company is a separate voting body and there is much variation in the distribution of voting shares among them. Moreover reasonable criteria exist against which to judge the indices. New algorithms for the Shapley-Shubik and Banzhaf indices are applied to detailed data on beneficial ownership of 444 large UK companies without majority control. Because some of the data is missing, both finite and oceanic games of shareholder voting are studied to overcome this problem. The results, judged against these criteria, are unfavorable to the Shapley-Shubik index and suggest that the Banzhaf index much better reflects the variations in the power of shareholders between companies as the weights of shareholder blocks vary

Strategic Behavior and Underpricing in Uniform Price Auctions: Evidence from Finnish Treasury Auctions

We study uniform price auctions using a dataset which includes individual bidders' demand schedules in Finnish Treasury auctions during the period 199299. Average underpricing amounts to .041% of face value. Theory suggests that underpricing may result from monopsonistic market power. We develop and test robust implications from this theory and find that it has little support in the data. For example, bidders' individual demand functions do not respond to increased competition in the manner predicted by the theory. We also present evidence that the Finnish Treasury acts strategically, taking into account the fact that the auctions are part of a repeated game between the Treasury and the primary dealers. Empirically, the main driver behind bidder behavior and underpricing is the volatility of bond returns. Since there is no evidence that bidders are risk averse, this suggests that private information and the winner's curse may play an important role in these auctions

Bidder Behaviour in Multiple Unit Auctions Evidence from Swedish Treasury Auctions

We analyze a unique data set on multiunit auctions, which contains the actual demand schedules of the bidders as well as the auction awards in over 400 Swedish Treasury auctions. First, we document that bidders vary their prices, bid dispersion and the quantity demanded in response to increased uncertainty at the time of bidding. Second, we find that bid shading can be explained by a winner's curse driven model where each bidder submits only one bid, despite the fact that the bidders in our data set use much richer bidding strategies. Third, we explore the extent to which the received theories of multiunit auctions are able to offer insights into the bidder behavior we observe. Our empirical evidence is consistent with some of the predictions of the models of auctions which emphasize private information, the winner's curse and the champion's plague. While the models of multiunit auctions serve as useful guideposts, our empirical findings also point to several new areas of research in multiunit auctions that are of policy and theoretical interest