Debates like these have apparently been raging on the Sony Pictures Jeopardy! board for years, although skimming over them it seems like the tone is slightly more pleasant over there (not that it's bad ITT).
Most of the time, the player in first (let's call him 1) bets enough to cover an all in bet by the player in second (let's call him 2). But should he? If player 3 is close enough to be a factor, 2 will often not bet all in but go small enough to protect against an all in from 3.
Here is a link to games satisfying the conditions of "Shore's Conjecture" which I won't go into rigorous detail about, but the gist of the conjecture is that if 2 has 3 locked out going into Final Jeopardy, 2 will bet "small" and keep 3 locked out, hoping 1 bets "big" (which usually happens).
The link has a review of seasons 20-25 of Jeopardy! showing that in games meeting those conditions, 2 in fact did bet "small" 93 out of 154 times (60.4%).
A guy's been keeping track of games satisfying the Shore conditions for seasons 20-25 and Shore (aka "small") betting has come out ahead of shutout betting 28-21, with 105 draws. That is, the Shore bet would have won 28 times when the shutout would have lost, the shutout bet would have won 21 times when the Shore bet would have lost, and 105 times it didn't make a difference.
As many have said, for most player 1s, betting small would have to be substantially better than shutout betting to offset the psychological trauma of being in first place, getting Final Jeopardy correct, and still losing.
Just thought I'd throw out some empirical data, I'm fully aware it doesn't take into account all the variables.