Optimal Strategies of Games

Contact us if you have a strategy question, whether it’s about an economic decision or about a game you enjoy playing, or read on for examples of how math can answer strategy questions.

Game theory is a subdiscipline of mathematics: the study of mathematical models of conflicts, to deduce what actions rational players will take.

The RAND corportation first issued Melvin Dresher’s popular primer on game theory to the public in 1961.

“Games” are defined very broadly. Game theory can be applied to card and board games where you seek to maximize your score or maximize your chance of winning. But it also applies to economics (buyers and sellers competing with one another for business), evolutionary biology (the “winners” pass their genes on to the next generation), and even (as infamously studied by the RAND Corporation in the 1950s and 1960s) war.

Game theory is not necessarily statistical, but many of the most interesting games to study are games of imperfect information, where players must act based on only a partial knowlege of the game conditions.

MarkAir was driven into bankruptcy in 1995 after a years-long fare war with Alaska Airlines.

The examples given below are couched in terms of recreational games or betting. But the same methods apply to business decisions, small and large. Should I fill my car’s gas tank before I leave town, even though the tank is half full, or should I drive 200 miles down the road until I am on Empty, gambling that the price will be lower there? Should an airline start a fare war in an effort to drive a rival into bankruptcy, even at risk of bankrupting itself? (When I was attending the University of Alaska Fairbanks, Alaska Airlines and MarkAir decided to play this game. MarkAir lost, after a 5-year battle.)

Simple betting strategy: the Kelly criterion

In 1956, John Kelly, Jr. of Bell Labs showed how to maximize one’s long-term profit betting on flips of a biased coin: if the coin is biased 51-49 in your favor, bet 2% of your bankroll at every toss; if it is biased 55-45 in your favor, bet 10% of your bankroll at every toss; 60-40 in your favor, 20%. (Read the Kelly Criterion article on wikipedia, or Kelly’s original paper, to see the proof.)

His basic method (maximize the expected value of the logarithm of your bankroll after each bet) can be extended to calculate a recommended bet size for any proposition where the probabilities and sizes of all the possible payoffs are known.

One interesting variation is the question of two simultaneous bets. Say you are offered even money bets on each of two football games played on the same day. You believe you have a 70% chance of correctly picking the winner of the first, and a 60% chance of correctly picking the winner of the second.

The simple Kelly criterion says to bet 40% of your bankroll on the first game. A conservative approach would be to consider the worst-case scenario of losing the first bet, and place 12% (20% of the 60% not already committed) on the second game.

But one can, with modern computing software, directly compute the ideal bet sizes: maximize

(.70)(.60) Log (1+x+y) + (.70)(.40) Log(1+x-y) + (.30)(.60) Log(1-x+y) + (.40)(.30) Log(1-x-y)

to find that betting 38.6% of your bankroll on the first game and 16.9% on the second is best, giving a slightly better expected rate of return (10.43% vs. 10.27%) than the conservative approach would.

Rock-paper-scissors and beyond

The rules of Rock-Paper-Scissors-Lizard-Spock.

In 2008, the TV show The Big Bang Theory popularized an expanded version of the childhood game: Rock Paper Scissors Lizard Spock (see the clip where Sheldon introduces Raj to the game.) In 2014, after watching this episode of the show together, my girlfriend commented that she couldn’t make the “Live Long and Prosper” sign from Star Trek. If she and I played this game, I would know that she would never throw Spock. She would have 4 moves to choose from while I would have 5. Obviously this gives me an advantage: but how much of one, and how do I play to take maximum advantage of the situation?

My manuscript “Rock-Paper-Scissors-Lizard-Spock against a handicapped opponent” is currently in press. A reprint will be posted here as soon as it is published.

Small problems inside larger games

I’ve spent a great deal of time studying problems that arise in my two favorite games, bridge and backgammon. Several of these analyses appear on my sister site, TaigaBridge, devoted to my bridge teaching activities:

• Backgammon:
  • An idealized cubeful racing game: how to handle the doubling cube in a simplified version of backgammon where only the total distance each player has left to travel, not the number of stones he has left, matters.
• Bridge:
  • A detailed analysis of a single declarer-play problem posed in Fred Gitelman’s training software, Bridge Master 2000 — coming soon.
  • Several articles discussing fine-tuning a bridge bidding system as a result of double-dummy simulations (repeatedly dealing out all the unseen cards, and seeing which of several actions is successful most often.)
  • How to choose the best action if you accidentally bar your partner from the bidding, forcing you to place the contract with a single bid instead of exchanging information with your partner