thieving professor

[acesfullmike 0]

so i love my professor, but he stole my bonus question off of another website. I need a little help here from the math guys, well here we go. Ok, since you guys like poker here is a game for you to play. Two players each put 1 unit into the pot. Each player gets a random number uniformly distributed between 0 and 1. Each player knows the value of his number but not the value of his opponent's number. The first player is then given an opportunity to bet one additional unit. If the first player doesn't bet there is a showdown and the player with the highest number collects the antes. If the first player bets the second player may call by matching the bet or drop out (giving the antes to the first player). If the second player calls there is again a showdown and the player with the highest number collects the pot (consisting of 4 units, the bets and the antes). If both players follow their optimal strategy what is the value of the game? In other words if they play (optimally) a large number of games how much is the first player expected to win (or lose if the value is negative) per game.At first I said the guy in position dominates this game. Then i realized, he doesnt get to bet when the first guy checks, whoops. The first guy wins this game, he bets when he is most likely ahead or likely to get paid from fold equity since we arent thinking about getting re-raised and he checks it down when he is most likely beat. So, we have to bet when we are exactly favored to win money. ex we have .501, time to bet since our opponent probably will call with .500 or better. But do we bet with .499, since our opponent may think our odds with what is in the pot already causes us to bet with .333 or better based on pot odds and what we are getting, oh my god im cross eyed. HELP.

[Sluggo 0]

Do you know how to calculate maximins and minimaxes?

[acesfullmike 0]

yeah, the problem i am having is figuring out exactly how much fold equity the first player has by betting. i mean player 2 knows that 1 only is going to bet when he feels like he is getting the best of it. but optimally, 2 has to call when he feels he is best only 25% of the time. so is 2 calling with anything better than .25. we know this isnt right since 1 would not raise the bet with .25 as he would not be best >=50%. i believe that 2 calls a raise only when he is going to be better than .5 in a 4:1 ratio. So i think 2's magic calling a raise number is .625. 1 wants a free showdown with anything less than .5 as putting more money in with less than the average would be foolish, unless his fold equity 'buys' him more ev. 2 has to call with anything that would possibly win 25% of the time due to pot odds. i am having trouble with this as it seems this is read dependant, but i dont think my prof will like that answer. can i answer, if 1s vpip is 75% call him down with anything over .25, but if his vpip is 50% only call down with .625 or better.

[acesfullmike 0]

good call on the maximins. wow game theory can make your brain hurt. its a von neumann poker model if anyone cares to check that out, pretty cool stuff, even proves that in the above situation that betting the pot is the best idea.