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Forming voting blocs and coalitions as a prisoner's dilemma: a possible theoretical explanation for political instability Gelman, Andrew Uploaded 10-27-2003 Keywords coalitions cooperation decisive vote elections legislatures prisoner's dilemma voting power Abstract Individuals in a committee can increase their voting power by forming coalitions. This behavior is shown here to yield a prisoner's dilemma, in which a subset of voters can increase their power, while reducing average voting power for the electorate as a whole. This is an unusual form of the prisoner's dilemma in that cooperation is the selfish act that hurts the larger group. Under a simple model, the privately optimal coalition size is approximately 1.4 times the square root of the number of voters. When voters' preferences are allowed to differ, coalitions form only if voters are approximately politically balanced. We propose a dynamic view of coalitions, in which groups of voters choose of their own free will to form and disband coalitions, in a continuing struggle to maintain their voting power. This is potentially an endogenous mechanism for political instability, even in a world where individuals' (probabilistic) preferences are fixed and known.

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Standard Voting Power Indexes Don't Work: An Empirical Analysis Gelman, Andrew Katz, Jonathan Bafumi, Joseph Uploaded 11-02-2002 Keywords Banzhaf index decisive vote elections electoral college Shapley value voting power Abstract Voting power indexes such as that of Banzhaf (1965) are derived, explicitly or implicitly, from the assumption that all votes are equally likely (i.e., random voting). That assumption can be generalized to hold that the probability of a vote being decisive in a jurisdiction with $n$ voters is proportional to $1/sqrt{n}$. We test---and reject---this hypothesis empirically, using data from several different U.S. and European elections. We find that the probability of a decisive vote is approximately proportional to $1/n$. The random voting model (or its generalization, the square-root rule) overestimates the probability of close elections in larger jurisdictions. As a result, classical voting power indexes make voters in large jurisdictions appear more powerful than they really are. The most important political implication of our result is that proportionally weighted voting systems (that is, each jurisdiction gets a number of votes proportional to $n$) are basically fair. This contradicts the claim in the voting power literature that weights should be approximately proportional to $sqrt{n}$.

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How much does a vote count? Voting power, coalitions, and the Electoral College Gelman, Andrew Katz, Jonathan Uploaded 05-08-2001 Keywords coalition decisive vote electoral college popular vote voting power Abstract In an election the probability that a single voter is decisive is affected by the electoral system -- that is, the rule for aggregating votes into a single outcome. Under the assumption that all votes are equally likely (i.e., random voting), we prove that the average probability of a vote being decisive is maximized under a popular-vote (or simple majority) rule and is lower under any coalition system, such as the U.S. Electoral College system, no matter how complicated. Forming a coalition increases the decisive vote probability for the voters within a coalition, but the aggregate effect of coalitions is to decrease the average decisiveness of the population of voters. We then review results on voting power in an electoral college system. Under the random voting assumption, it is well known that the voters with the highest probability of decisiveness are those in large states. However, we show using empirical estimates of the closeness of historical U.S. Presidential elections that voters in small states have been advantaged because the random voting model overestimates the frequencies of close elections in the larger states. Finally, we estimate the average probability of decisiveness for all U.S. Presidential elections from 1960 to 2000 under three possible electoral systems: popular vote, electoral vote, and winner-take-all within Congressional districts. We find that the average probability of decisiveness is about the same under all three systems.

4

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Estimating the Probability of Events That have Never Occurred: When Does Your Vote Matter? Gelman, Andrew King, Gary Boscardin, John Uploaded 10-27-1997 Keywords conditional probability decision analysis elections electoral campaigning forecasting political science presidential elections rare events rational choice subjective probability voting power Abstract Researchers sometimes argue that statisticians have little to contribute when few realizations of the process being estimated are observed. We show that this argument is incorrect even in the extreme situation of estimating the probabilities of events so rare that they have never occurred. We show how statistical forecasting models allow us to use empirical data to improve inferences about the probabilities of these events. Our application is estimating the probability that your vote will be decisive in a U.S. presidential election, a problem that has been studied by political scientists for more than two decades. The exact value of this probability is of only minor interest, but the number has important implications for understanding the optimal allocation of campaign resources, whether states and voter groups receive their fair share of attention from prospective presidents, and how formal ``rational choice'' models of voter behavior might be able to explain why people vote at all. We show how the probability of a decisive vote can be estimated empirically from state-level forecasts of the presidential election and illustrate with the example of 1992. Based on generalizations of standard political science forecasting models, we estimate the (prospective) probability of a single vote being decisive as about 1 in 10 million for close national elections such as 1992, varying by about a factor of 10 among states. Our results support the argument that subjective probabilities of many types are best obtained via empirically-based statistical prediction models rather than solely mathematical reasoning. We discuss the implications of our findings for the types of decision analyses that are used in public choice studies.

5

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Estimating the Probability of Events That have Never Occurred: When Does Your Vote Matter? Gelman, Andrew King, Gary Boscardin, John Uploaded 02-14-1997 Keywords conditional probability decision analysis elections electoral campaigning forecasting political science presidential elections rare events rational choice subjective probability voting power Abstract Researchers sometimes argue that statisticians have little to contribute when few realizations of the process being estimated are observed. We show that this argument is incorrect even in the extreme situation of estimating the probabilities of events so rare that they have never occurred. We show how statistical forecasting models allow us to use empirical data to improve inferences about the probabilities of these events. Our application is estimating the probability that your vote will be decisive in a U.S. presidential election, a problem that has been studied by researchers in political science for more than two decades. The exact value of this probability is of only minor interest, but the number has important implications for understanding the optimal allocation of campaign resources, whether states and voter groups receive their fair share of attention from prospective presidents, and how formal ``rational choice'' models of voter behavior might be able to explain why people vote at all. We show how the probability of a decisive vote can be estimated empirically from state-level forecasts of the presidential election and illustrate with the example of 1992. Based on generalizations of standard political science forecasting models, we estimate the (prospective) probability of a single vote being decisive as about 1 in 10 million for close national elections such as 1992, varying by about a factor of 10 among states. Our results support the argument that subjective probabilities of many types are best obtained via empirically-based statistical prediction models rather than solely mathematical reasoning. We discuss the implications of our findings for the types of decision analyses that are used in public choice studies.