Video checking of referee decisions

Good day, sports fans. Great Lakes Economics brings you a different dish from our usual fare, consisting of a probabilistic analysis of the recent introduction of video and "third umpire" checking of the decisions of referees or umpires. It's sports news, analysed, and I conclude that under quite general conditions, weaker players and bowlers are disfavoured by the change. Innings lengths are increased.

Suppose that a referee decision has happened. Write E* to be the event that the referee judges that something occurred on the pitch such as a goal or wicket, and E to be the event that the something really did happen. Then by conditional probability,

P(E*) = P(E*¦E).P(E) + P(E*¦not E).P(not E)

where P(E*) denotes the probability of E*, P(E*¦E) and so on denotes the probability of E* given that E has occurred, and "not E" means E didn't happen. We assume that the referee makes an error with a probability q, that is, P(E*¦not E)=q and P(E*¦E)=1-q. We abbreviate P(E)=p, so P(not E)=1-p. The equation becomes

P(E*) = (1-q).p + q.(1-p) = p + q.(1-2p)

Under checking of referee or umpire decisions, we assume that the assessments are accurate, so that the probability of P(E*¦checked)=P(E). The difference P(E*¦unchecked)-P(E*¦checked) is equal to q.(1-2p) which is greater than zero if p is less than 0.5 (which it presumably is, since referees and umpires are often appealed to by players without their entreaties being heard). Moreover, p is smaller for weaker teams than stronger teams since they are less likely to convert marginal chances, so the difference for them will be larger and they have more of an advantage from referee errors. It might be why managers of weaker teams are less prominent than managers of leading teams in their criticisms of referees, at least in the UK. We may also view bowlers as having a lower p than batters in cricket as they often have near misses from LBW or loose edges, so they are likely to be disadvantaged relative to batters by the new rules. Innings lengths are likely to be increased, as innings are more skill based - batters do not have to be both skilful and lucky (at least in umpire rulings) to get big scores.