Marginalism, Tournaments, and Extra Effort

The theory of tournaments and economic theory in general provide an explanation for why tournament winners (e.g. in golf, tennis, beauty contests, or many olympic events [via endorsements, etc.]) receive so much more prize money than second place finishers. Clearly the productivity or talent or skill or beauty of the winner is not much greater than that of the runner-up. So the differentials in the prizes do not reflect any of the usual marginal productivity determinants of compensation.

And yet, if you were running a competitive tournament (other than auto racing and the like), you would want big prizes for first place finishers. Similarly, if you are a stockholder in a major corporation you want much bigger salaries for the CEOs than for the senior VPs. You want to hold out big prizes for the winners so that all the contestants will put forth LOTS of effort to win: if the extra compensation for winning were not very large, then there is a reasonable chance that contestants would not give that "extra effort" to try to win -- they would be more likely to be content to settle for second-place winnings (and the attendant lower levels of effort), and your tournament (or corporation) would not be as good.

If this is true, we should see many more attempts at spectacular plays in do-or-die situations (e.g. if a team will be eliminated from making or staying in the play-offs if it loses; or if golfers are on the last few holes of a major tournament) than during the regular season. During the regular season, it makes far less sense to risk injury by making a diving catch in baseball or a lunge that might land you awkwardly in some other sport; in eco-speak, the marginal benefit of the spectacular play is outweighed by the expected marginal cost, including the risk of injury. Likewise, in golf, if you're trailing by one shot or tied near the end of the tournament, the extra benefits of making a spectacular shot might very well be greater than the costs of trying it and failing; earlier in the tournament you would have more incentive to play safer shots to try to stay close.

During playoffs, especially when your team (or you) are facing elimination, the expected incremental benefits of making a spectacular but dangerous play might very well outweigh the expected incremental costs

Are there any specific tests of this hypothesis? How about comparing numbers and/or seriousness of injuries in these games versus during the regular season? Or what about the number of golf shots in the water in these situations vs. earlier in the match?