In Chapter 5, where we explored the law of averages, binomial distributions, and the truth about betting systems, we learned an important fact: “In a game with unchanging probabilities, no strategy can change the eventual expectation.” Now we need to turn the question around: “In a game with opponents, how should we strategize?”
“Chapter 6 of The Mathematics of Games and Gambling answers this very question.
This time, we’re going beyond the world of probability and into the world of game theory. It’s a tool for mathematically analyzing how your choices should be different in the presence of an opponent.
More than just a gambling technique, it’s a theory that has influenced modern economics, political science, and even artificial intelligence research. Its impact has been enormous. In fact, in 1994, John Nash, John Harsanyi, and Reinhard Selten were awarded the Nobel Prize in Economics for their work on game theory analysis. With casino math and game theory out of the way, it’s time to move on to the math of strategy.
Theory Part: Bargaining or Betrayal?
The Prisoner’s Dilemma, a casino version of a human psychology experiment
There’s a very famous story in the book to illustrate game theory. It’s the “prisoner’s dilemma”: two people commit a robbery together and get caught. When they’re being investigated separately, they have two choices
- If they both confess: Both medium (-4)
- If both are silent: Short sentence for simple possession of a weapon (-1)
- If only one confesses: The confessor is released (10), the silent one is guilty of everything (-6).
Mathematically, confessing is the dominant strategy. Whether your opponent stays silent or confesses, you’re better off confessing, so you end up with both confessing and being sentenced to -4. But is this best?
No, because if we’re both silent, it’s only -1. But if you’re afraid the other person will confess, you confess, and that’s the dilemma.
Poker and the Prisoner’s Dilemma
This structure is also seen frequently in poker.
For example, let’s say you and your opponent both have a strong hand, and you’re trying to decide whether to bluff, call, or fold.
- If neither of you bluff, you’re safely playing for a small pot.
- If one player bluffs and the other folds, the bluffer has a big advantage.
- What if they both bluff? Someone loses big.
This creates a situation where if you can’t trust each other, you have to bluff, and the strategy of bluffing becomes the dominant strategy, even if it costs you. It’s like a prisoner’s dilemma structure.
Strategies for iterative games and interactions
The book explains that cooperation becomes possible if the prisoner’s dilemma is repeated many times. Because you can see whether your opponent is betraying you and retaliate in the next round, cooperation can be sustained for the long term. It’s an “eye for an eye, tooth for a tooth” strategy. In poker, memorizing the frequency of your opponent’s bluffs and responding to them is a practical application of this iterative game strategy: if they cheat you, you retaliate in the next hand, and if they cooperate, you cooperate. But that’s in an open-ended iterative game, where you don’t know when it’s going to end.
What if we look at a recurring game with a clearly defined number of rounds, say a game that ends after 15 rounds?
- Since there is no retaliation from the opponent in the final game, most players choose to betray.
- If your opponent realizes this, they will consider betraying you in the previous game…
- Working backwards in this way eventually leads to betrayal being the only rational choice all along.
This is called backward induction, a logic that shows that in an iterative game with a fixed end, cooperation is bound to break down.
In this way, game theory explains that the likelihood of cooperation depends on the structure of the iteration.
Scalability in Game Theory: Politics and AI
1) Majority Gaming and the U.S. Congress
Game theory isn’t just for gambling; it’s also used to analyze political systems.
For example, when analyzing the structure of bill passage in the U.S. Senate, House, and President, we use a concept called the weighted voting game.
A weighted voting game is a game in which each voter is given a different vote (weight)and must exceed a certain threshold (quota) to win. The U.S. legislative process can be analogized to this game.
- Players: House of Representatives (435), Senate (100), President (1)
- Weight: Number of votes for each (435, 100, 1)
- Win condition: All three players – House, Senate, and President – must agree to pass the bill (i.e. win)
It’s important to note that this is a conjunction game, meaning that all players must agree to win. No matter how unanimous the 435 members of the House of Representatives and 100 members of the Senate, if the President vetoes a bill, it will not pass. Because of this structure, the president’s vote doesn’t just carry 1/536 (435+100+1) weight, it actually carries about 1/270 weight.
2) How does AI learn to play games?
The book also shows an interesting example of game theory. It’s about how computers learn to play games – AI meets game theory.
Computer Learning with the HEXPAWN Game
The book introduces HEXPAWN, a simple chess-like game. It’s a very simple game where you move three pieces on a 3×3 board,
- Analyze wins and losses in a tree structure,
- For each node, record the best number,
- It learns as it goes along, avoiding incorrect numbers.
In this way, the computer can construct a strategy that “learns as it loses,” and eventually become a perfect player.
Unlike HEXPAWN, chess is so complicated that you’d have to solve 10^75 cases just to compute 25 numbers.
Because of this complexity, computers use simplified strategy evaluation criteria called heuristics instead of full tree computation.
As a result, computers began to beat humans at games like checkers, chess, and Go, and game theory was extended to learning models for AI.
Humans can’t do as many probability calculations as fast as computers, which is why being able to calculate probability is so important to winning and losing.
FAQ: Questions about game theory and gambling
If it’s a gambling game with opponents, of course. It becomes the core theory in poker, bluffing, negotiation, heads-up games, and so on.
If decisions like bluffing, calling, or folding are structured in relation to an opponent’s choices, then a strategy selection similar to the Prisoner’s Dilemma is required. Repeatedly discerning an opponent’s style and responding to it is also an application of game theory.
This enables psychological warfare along the lines of “If you betray me this round, I’ll betray you next round,” thereby sustaining cooperation.
Game theory makes your strategy ‘unpredictable’. For example, imagine you hold both a ‘truly strong hand’ and a ‘bad hand for bluffing’ during the final betting round in poker. Game theory then mathematically calculates the ‘frequency of betting with the real good hand’ and the ‘frequency of bluffing with the bad hand’. Using this mixed strategy makes it impossible for your opponent to predict whether your bet stems from a strong hand or a bluff.
Yes, there is. According to game theory, bluffing isn’t done recklessly. You must calculate a specific frequency to bluff that forces your opponent to either call or fold indiscriminately. For example, this is why you need to adjust your bluffing frequency as your opponent’s bets increase and the pot grows larger. This becomes the core principle of achieving ‘bluffing balance’.
Finalize
Game theory is not just a branch of mathematics, but a powerful analytical tool that encompasses human psychology, strategy, cooperation, and even betrayal. In my next post, I’ll introduce Chapter 7, the final chapter of The Mathematics of Games and Gambling, where we’ll analyze various gambling cases with math, including lotteries, horse racing, and bankruptcy theory. Let’s dig into the final secrets of casino math together.
