## Saturday, December 29, 2007

### Game Theory in Las Vegas

I tried to use a bit of game theory to increase my odds when gambling in Vegas. Of course, I already know that the casinos have a house advantage so I wasn't surprised to see negative expected utilities when calculating the odds of a game. The rational choice would of course be not to gamble. Instead I just gambled a little.

After doing a bit of research on Wikipedia, some of the games with the worst odds are Keno and slot machines. However, these are the games I ended up playing. The games with better odds such as blackjack and roulettes had much higher minimum bets (especially compared to the one cent slots). Games which involve skill allow you minimize the house advantage.

Keno was slightly more interesting than slots because each casino had different paytables and specialty bets. You can choose bets with different payoffs and different probabilities of winning. I could calculate when one type of bet had a lower expected utility than another.

My brother came into Vegas with his own gambling strategy. Given a game with fifty-fifty odds and double payoff, his strategy was to continue doubling his bet until he wins. In the end, if he had an infinite amount of money to bet, this guarantees that he will win exactly the value of the first bet. For example: lose \$1, lose \$2, lose \$4, win \$8, profit=\$1. However, with a finite amount of money to bet I tried to persuade him that the strategy does not make any sense. It is risking losing lots of money (with a low probability) and gaining a little (with a high probability). In the end this does not change your expected utility (which is zero for the game I described above). The strategy also doesn't make sense because of the concept of sunk cost. Once he loses money in previous games, it should not affect his future betting strategy. Despite my complaints he ended up making a profit o_o

Kevin Leyton-Brown gave an interesting guest lecture about game theory in my Cognitive Systems 300 class last year. Near the end of class we had several different types of auctions for \$5 bills. He let us know that any profits would go to charity. I thought one of the auctions was particularly interesting in which the highest bidder would win the prize as normal but the second highest bidder would also have to pay their bid. This means as soon as there was more than two bids over \$2.50 Kevin was guaranteed to make a profit on the \$5 bill. I made bids over \$2.50 even though I knew it wasn't an optimal strategy (it was for charity after all). Being the second highest bidder you are really tempted to outbid the highest in an attempt to win the prize and avoid a complete loss. Another guy in the class and I continued bidding against eachother until the bids actually got over \$5. This means Kevin is getting over \$10 for his \$5 bill :). I eventually gave up and ended up being the second highest bidder. The highest bidder only lost a few cents while I lost over \$5. I thought the auction was fascinating due to the temptation for the two highest bidders to continue outbidding eachother despite going over the actual value of the prize (let alone the optimal maximum of \$2.50). I'll keep this auction in mind as a fun thing to try with my friends (except this time I will be the auctioneer).