# Notes

Gibbons (1992), Osborne (2009) and Harrington (2009) are useful references on game theory. The analysis of the mixed strategy equilibria in the crime-reporting game draws on the classic paper of Palfrey and Rosenthal (1984), who examine the general issue of contributions to a discrete public good when contributions can and cannot be refunded. The model of strategic evidence gathering presented in this chapter is due to Skaperdas and Vaidya (2011). The examination of biased judicial decision making is a modified version of Farber (1980). Thorough analyses of preferences aggregation procedures and paradoxes can be found in Mueller (2003) and Saari (1995, 2001a, 2001b). Austen-Smith and Banks (2000, 2005) have an advanced treatment.

# Appendix

This appendix proves the Condorcet Jury Theorem stated in the text. Each juror's decision can be characterised as a Bernoulli random variable *X,* where

The distribution of *Xj* is as follows:

Let *S _{n}* = ^ X be the number of jurors who make the correct decision,

*j=1*

remembering that juror decisions are statistically independent. The event that at

*n* + 1

least half of the jurors will make the correct decision is the event that *S _{n} >*-.

- 1 We want to show that if
*p >*, then: - 2

To this end, note that:

1 - Pr *(n - S _{n} >* j = 1 - Pr

^{(}

*S*j = 1 - Pr (

_{n}< n -*S*j = Pr (

_{n}<^{n}^^{1}*S*j

_{n}>So, to show the result, it suffices to show that li_{m}P_{r} | _{n -} S > ^{n}—-1 = 0. We use

l. " 2 ^{J}

the one-sided Chebyshev inequality [see, for example, Ross (2010) page 403], which states that for any random variable Y with mean zero and variance *a*, for any positive number *a,* we have

Now the mean of *n - S _{n} - n + np=* 0, and its variance is np(1 - p). Hence we have

^{1}

48 *Law and Markets*

1

where the strict inequality follows from the assumption that * ^{p}>* 2 . The last expression goes to zero as

*n ^*<» .