In the top tennis tournaments there is a limited instant-replay system. When a player disagrees with a call (or non-call) made by a linesman, he can request an instant-replay review. The system is limited because the players begin with a fixed number of challenges and every incorrect challenge deducts one from that number. As a result there is a lot of strategy involved in deciding when to make a challenge.
Alongside the challenge system is a vestige of the old review system where the chair umpire can unilaterally over-rule a call made by the linesman. These over-rules must come immediately and so they always precede the players’ decision whether to challenge, and this adds to the strategic element.
Suppose that A’s shot lands close to B’s baseline, the ball is called in by the linesman but this call is over-ruled by the chair umpire. In these scenarios, in practice, it is almost automatic that A will challenge the over-ruled call. That is, A asks for an instant-replay hoping it will show that the ball was indeed in.
This seems logical. It looked in to the linesman and that is good information that it was actually in. For example, compare this scenario to the one in which the ball was called out by the linesman and that call was not over-ruled. In that alternative scenario, one party sees the the ball out and no party is claiming to see the ball in. In the scenario with the over-rule, there are two opposing views. This would seem to make it more likely that the ball was indeed in.
But this is a mistake. The chair umpire knows when he makes the over-rule that the linesman saw it in. He factors that information in when deciding whether to over-rule. His willingness to over-rule shows that his information is especially strong: strong enough to over-ride an opposing view. And this is further reinforced by the challenge system because the umpire looks very bad if he over-rules and a challenge shows he is wrong.
I am willing to bet that the data would show challenges of over-ruled calls are far less likely to be successful than the average challenge.
A separate observation. The challenge system is only in place on the show courts. Most matches are played on courts that are not equipped for it. I would bet that we could see statistically how the challenge system distorts calls by the linesmen and over-rules by the chair umpire by comparing calls on and off the show courts.
Cheap Talk 
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January 24, 2010 at 11:27 pm
afinetheorem
Jeff,
I’m sure you’re also aware how this has played out in the NFL. Some plays are reviewable – a positive ruling of a fumble which allows play to continue until a recovery – and some are less so – a negative ruling of a fumble which stops play as soon as the player is whistled down. As far as I know, there is in fact evidence that referees err on the side of caution when replay is available. That is, they allow play to continue on potential fumbles more often than if there was no review. In this case, the fact that review exists should make you more willing to ask for a replay, not less.
The tennis example is interesting because there are two chances to overturn the linesman. I imagine the analysis above is correct – if the chair overturns a call, she had better be sure she was right and not the lineman. However, the Nash solution is tricky here. If the chair’s “overturn function” is a function of confidence in correctness and disutility from being proved wrong, then her willingness to overturn is decreasing in the probability that the player will challenge her call. If players always challenge, the chair will rarely overturn. However, players know this, and will challenge less. Because of this, the chair will overturn more! I’d imagine the overall effect is ambiguous, but my hunch is that, conditional on the challenge rate of a player overall (that is, on points where the chair does not overturn), a chair will overturn more often.
January 27, 2010 at 2:01 pm
Anonymous
I think you also have to consider a few other factors:
– In tennis, not all points and situations are equally important. Consequently players who actually try to strategize their challenges consider the score and remaining challenges before challenging. Thus if player A is Ivo Karlovic, and the overrule happens at 40 love on his serve, while he still has all three of his challenges, he’ll probably shrug it off and hit an ace on the next point (which is what happens on close to half his serves). He’d probably do the opposite if the overrule resulted in a break, or took an opportunity to go up on his opponents serve away from him.
– Different umpires have very reputations among players for fairness and correctness. Also, certain players believe that certain umpires have it in for them regardless of the reputation of those umpires among players on the whole. I don’t know of any current specifics as dramatic as, say, Jeff Tarango/Bruno Rebeuh, but I bet, say, Andy Roddick (who is very aware of who officiates his matches and gets into a lot of arguments with umps) considers who’s in the chair before challenging.
– Players have very different attitudes towards the challenge system. For instance, Roger Federer is quite vocal in his dislike of the whole thing, and seems to take perverse pride in making spurious challenges. Given that he plays more matches on show courts than anyone else in the sport, that alone has to skew the statistics.
– Some players argue, some don’t.
I think it would be very hard to control for these variables. I also question your statement about challenges of overrules being almost automatic in the scenario you posit, given all the other noise surrounding player pychology. It would be interesting to see the analysis, but I have a hard time imagining it being convincing.
January 28, 2010 at 11:45 am
Nitesh Garg
I always wonder about the accuracy of the instant replay system which shows the mark of the ball as completely round and a big as the ball itself. But actually when a round object would hit ground only a small part would actually touch the ground. How fair are those overrules when the instant replay shows the ball to be in or out by very small margins?