I have been listening to More or Less for some time, one of the few programmes that challenges the use and abuse of statistics in politics and media. I tend to listen to the podcasts, so I have only just caught up with some of the broadcasts from January, and I'm joining a debate that has already run a number of rounds.
In a discussion of the current crisis in banking and bankers' bonuses, there was a claim that bankers are better rewarded by following the same strategy as everyone else in the bank, rather than trying to be different. The logic suggests that if one banker follows an alternative strategy (as described in the show), then he will be more successful than others (perhaps) 75% of the time and less successful 25% of the time. Assuming that the "normal" strategy is followed by all other bankers and this is successful only 50% of the time, then the alternative strategy will be less successful overall, as he will only generate personal success 75% of the times when the bank is successful overall (rather than 100% following the "normal" strategy).
But it struck me that this analysis is only looking at one aspect of the probability - the correlation between individual success and overall success. And, while it is true that the alternative strategy will only be partially correlated with bank success, while the normal strategy will be fully correlated with overall bank success, you also need to consider the quantity of bonus paid.
Claiming greater success for the "normal" strategy assumes that the same bonus is paid to all successful bankers when the bank does well – it does not account for the differential bonus that the most successful bankers will earn. For example, if the most successful banker earns twice the bonus of the others, then the alternative strategy will generate 50% more bonus on average.
The value of the alternative strategy depends on the level of extra bonus for the most successful people. At 1.33 (= 1/(75%)) times the average, the alternative strategy is equal to the other strategy, but for any more bonus, the alternative strategy is clearly superior. I have modelled this in a spreadsheet.
So the suggestion from the show is incorrect - there is an incentive for bankers to break from the "normal" strategy. The challenge for the banks is how much to reward the most successful with bonuses to encourage alternative strategies ... which seems to have been the cause of the problem as well ...
Apologies if this makes little sense - you probably need to hear the show to understand the argument. I have submitted this post to the radio show team for comment - we'll see what they make of it.