Disclaimer. Don't rely on these old notes in lieu of reading the literature, but they can jog your memory. As a grad student long ago, my peers and I collaborated to write and exchange summaries of political science research. I posted them to a wiki-style website. "Wikisum" is now dead but archived here. I cannot vouch for these notes' accuracy, nor can I even say who wrote them. If you have more recent summaries to add to this collection, send them my way I guess. Sorry for the ads; they cover the costs of keeping this online.
Palfrey. 2005. Laboratory experiments in political economy.
Palfrey reviews a few recent experiments relative to the turnout paradox. He adopts the game theoretic perspective of Palfrey and Rosenthal (1983): Turnout is a "participation game," and zero turnout is not an equilibrium solution. Turnout is higher when elections are closer, and the minority party turns out at a higher rate than the majority party.
S&S tried to test the Palfrey-Rosenthal theory by running winner-take-all (W) and proportional representation (PR) rules in two-party experiments involving 12, 14, or 28 "voters" in each election (thus, a 2 x 3 design). Their findings are interesting: Turnout starts at around 50% in the W elections but declines (with repetition) to around 20%, but turnout starts at only 30% in the PR elections, also declining to around 20%. This study found that electorate size and party size are negligible.
Palfrey comments that this study was poorly thought through. The Palfrey-Rosenthal theory identifies multiple equilibria, which would present serious strategic ambiguity to the experiment's participants as designed. Thus, it's hard to know what to make of this experiment.
L&P build on a different model, Palfrey and Rosenthal (1985), which assumes that all members of a party reap an equal benefit from victory, but they face varying (privately known) costs of voting. The equilibria here involves cutpoint strategies--voters with a cost below some critical point vote, others abstain. For any given set of parameters, there is a unique Bayesian Nash equilibrium.
L&P have four experimental electorate sizes (3, 9, 27, or 51 voters) and two conditions: toss-up (where the two parties are almost equal in size) and landslide (where two-thirds of voters belong to the same party). Thus, a 4 x 2 design (well, sort of: the toss-up and landslide conditions are identical for the three-voter case).
The experiment validates the theory. Turnout declines with N (electorate size) and increases in the turnout condition. The minority turns out at a higher rate than the majority in the landslide condition
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