# Strategic evidence gathering

Consider the previous Bayesian updating framework, and suppose that there are two parties - a plaintiff and a defendant - to a legal dispute. Suppose that they gather evidence and produce to the court the evidence pair *E = (e*_{P}, *e*_{D}). The cost of obtaining *e* 'units' of evidence is simply assumed be $*e*. Suppose that:

The ratio ^{p(ep,e}o I ^{G)} is called the *likelihood ratio,* and is defined as P(*eP*,*eD* 1*1*)

the ratio of the probability of observing the evidence pair *E = (e*_{P}, *e*_{D}) when the individual is actually guilty to the probability of observing the evidence pair *E = (e*_{P}, *e*_{D}) when the individual is innocent. Note that the likelihood ratio can take on any positive value.

The assumption in equation (2.6) simply states that this ratio is proportional to the ratio of expenditure on the production of evidence by

*e*

each party. If this is the case, then p(*E* | *G*) = p(*E* 1*1*) -, and:

^{e}D

If the court has *uniform* prior beliefs, so that *p* (G) = *p* (*I*), then this collapses to:

Equation (2.8) states that if the court is an efficient processor of information and has uniform prior beliefs, and if the likelihood ratio is proportional to the ratio of expenditure on the production of evidence by each party, then the party that spends more resources on evidence production at trial is more likely to win the case, irrespective of whether the defendant is actually guilty or not. On the other hand, if the court has non-uniform priors and instead has some predetermined bias of guilt [so that *p(G*) ф p(I)], then the probability that the plaintiff wins is given by equation (2.7) above.

Even though courts in this example are efficient processors of information, the probability of either outcome occurring depends on the evidence-gathering ability of the two sides in the legal contest and is influenced by the way the evidence-gathering game is set up. Suppose that the marginal cost of producing evidence is 1 for each party. If the judgement has a value of $/ to the plaintiff, and the defendant stands to lose $/ if the plaintiff is successful, and if they are both risk neutral, then the expected net benefits to the plaintiff are:

and

for the defendant. Question 2 at the end of this chapter asks you to find the equilibrium of this game.