Three-way ties and Jeopardy: Or, Drew questions the odds 

 It's been in the news that a three-way tie happened on Jeopardy on Friday night. From the  AP article : 

 The show contacted a mathematician who calculated the odds of such a three-way tie happening — one in 25 million. 

 I have to believe that the mathematician contacted didn't have all the facts (and the AP rushed to meet deadline), because once you're in Final Jeopardy there's little randomness about it. It's all down to game theory. 

 Suppose we first estimate the odds that all three players are tied at the end of Double Jeopardy.The total dollar value shared by all three is around $30000, or about $10000 each. Since questions have dollar values which are multiples of $200, we could reasonably assume that there are 100 dollar values, between 0 and 20000, where each player can end up. So the odds of a tie at this stage should be no more than one in a million - and this is a very conservative guess, since I assume that the probabilities are all equal (whereas they would likely have a central mode around 10000.) 


 Breaking a three way tie with a Final Jeopardy question would then require that all three players bet the same amount, and I think the odds are considerably less than 1 in 20 that they'd all bet the farm no matter the category. 

 But it shouldn't even get that far. The scenario on Friday night had two players tied behind the leader who didn't have a runaway. So we have somewhere around 1 in 20,000 odds that this would happen (the factor of two because the third player could be ahead or behind the tied players.) 

 The runners-up would both be highly likely to bet everything in order to get past the leader. And the leader, in this case, placed a tying bet for great strategic reasons - getting one more day against known opposition rather than taking the chance of a new superstar appearing the next day - as well as a true demonstration of giving away someone else's money to appear magnanimous. 

 Even if the leader only had a 10% chance of making that call, and given that the other two players were pressured to bet high, that's still 1 in 200,000 - over 100 times more likely with a fairly conservative estimation process.