Tuesday, October 18, 2011 Game Recap & Discussion [SPOILERS]

[Estock82]

Posting here because you are discussing wagering and cant figure out how to start new thread.

Is there any discussion of optimal wagering according to game theory in a 2 person final jeopardy? I see a discussion of "break points" on the j-archive. I am interested in calculating both the expected value and optimal strategy in each zone between the break points. Most of the strategy options offered / suggested on the board are exploitive (not optimal) strategies, based on exploiting opponents tendencies in certain situations to make exploitable plays.

So far I have.

When the trailing player has less than 1/2 of the leading player, the leading player should bet between 0 and his maximum safe bet. His expected value of winning is 100%.

Between 1/2 and 2/3. The leader should bet enough to put the game out of reach. His expected value of this position, assuming the trailing player wagers correctly, is 75% wins.

Between 2/3 and 3/4. This is the point at which balanced wagering strategies come into play. Imagine that the game will be played repeatedly from this situation. Any one strategy could be exploited by the other player. The leader could bet enough to put the game out of reach, but then the trailer could bet to win on a double stumper as well as when the leader misses (by betting 0 or some small amount that wins on a double stumper) and gain a winning percentage of 50%. The leader could counter by betting zero and then winning 100% of the time that the trailing player makes the small bet. This is why a balanced strategy is necessary. If the leader balances his strategy by betting to put the game out of reach 2/3 of the time, and betting nothing 1/3 of the time, he will guarantee himself a winning percentage of 66.67 no matter what strategy the trailing player employs. In practice the leading player plays exploitively based on the knowledge that the trailing player goes "all in" too frequently. (the trailing player should balance his strategy with 2/3 large bets and 1/3 betting nothing, securing for himself his 33% expected value in this situation. Strategy advice in the j-archive suggests the trailing player to play exploitively by betting small or zero and increasing his winning percentage from 25% to 50%. In practice this is good advice.

So for the 2/3 to 3/4 range the expected value for the winning player is a 66.67 winning percentage by randomizing his betting strategy to include 2/3 bets (out of reach) and 1/3 zero bets.

I also calculated the EV (winning percentage) of the leading player when the trailer has 3/4 to 4/5, but I am not sure of all of the wagering possibilities and may have made an error. What I came up with suggests that optimal strategy for the leading player is to balance betting enough to put the game out of reach, with another bet which is enough to cover the trailing players maximum total when the trailing playing wagers to win on a DS. If he makes these bets with equal frequency, his expected value of this position is 62.5%. The trailing player should also employ 2 strategies in equal frequency, going all in, and making a bet that will win for him on a double stumper. He will win 37.5% of the time.

That is as far as I have gotten. There could be wagering strategies I failed to consider in the above example. I am having trouble calculating the EV for the 4/5 and above total for the trailing player. In this case, the leading player can not make a suitable small bet to protect against the trailers DS protective bet, without also wagering enough to fall below a zero wager by the trailing player. There seem to be many more options and it is not easy to recommend an optimal betting strategy.

If anyone has any feedback or insight on this and other situations, I would appreciate it.

[Estock82]

I'm making a few assumptions, such as a 50% correct response to final jeopardy. Assuming that easy questions offer a more likely double get and that tough questions offer a more probably double stump further complicates the question. Also, ignoring the monetary value of the final score and assuming that each player plays for the chance to come back as champion.

[Beginner's_Luck_500]

I was quite surprised that there were two graduate students who were the challengers, facing against a business manager, who was the champion. Throughout the game, I didn't quite as much knew one thought after another in those clues that came up. On the 'Remember 2008?' category, I accidentally thought of Wells Fargo on the $800 clue, but I didn't know that AIG came up in a different category, and the FJ! category was one of the hardest for me, because I obviously had nothing come into mind.
But now I know for one thing, Alexander Graham Bell was born in Scotland.