I recently picked up a book at the library:
Gladiators, Pirates and Games of Trust by Haim Shapira
The book covers some interesting concepts and problems in game theory, and how these scenarios play out in real-life negotations. It does not go into a lot of detailed explanation, but it’s a great starter for the topic. A host of problems are covered in this book, including: The Prisoner’s Dilemma, Stable Marriage Problem and Gale-Shapley algorithm, the irrelevance of order in the Gladiator game, and Evolutionary Stable Strategies.
I’ve written down a brief of the problems that intrigued me below.
Nash Equilibrium
A Nash Equilibrium is a set of strategies such that players will not regret their decisions once the other strategies are revealed. It’s interesting that these equilibria are not necessarily the set of strategies that give players the most optimal outcomes.
In the case of the Prisoner’s Dilemma, the Nash Equilibrium is for individuals to make the egotistical decision to betray their opponent. The net effect for both players is strategically unwise, and both players were better off if they collectively decided to play against their own interest.
The Blackmailer Paradox
In the Blackmailer Paradox, two players have to agree on how they should split a sum of money (within a time limit). Else, both players get nothing.
Just by playing the game, players have already won. The collective effect on both players is that they never lose any money. However, players might be more willing to “prove a point” and walk away with nothing than to walk away with the short end of the stick.
Rational players should agree to split halfway. This is the Nash Equilibrium. Yet, players may be irrational, going for an all-or-nothing scenario (give me 90% or I’d rather walk away with nothing!). When playing against an irrational player, it’s more optimal to act irrationally as well.
Reminds me a little of multiplying two negatives to get a positive.
Ultimatum Game
The Ultimatum Game is a variant of the Blackmailer Game, where one of the players proposes the split and the other player merely responds. There is a limited number of proposals that can be made.
An interesting effect is that responders who know the total amount being split might play irrationally. As a responder, you will gladly accept $100, but upon knowing that the total was $10000, you might feel indignant and reject the offer.
Chomp
Chopmp is a famous chocolate-eating game, where the opening player is known to have a winning strategy. However, the proof-of-existence is non-constructive (i.e. this strategy is not specified). A similar experiment would be the game of Nim.
I recall being fascinated by this game in middle school, where we’d crack our heads at this site. No matter how much I played, I could not outsmart the AI.
Apn idea struck that I should let the AI play against itself. I’d chomp the corner square and let the AI have the next turn. Then, no matter how the AI moves, it would be possible to replicate both my move and the AI’s move in one turn. I’d then start another game session with that move, effectively letting the AI respond to its own algorithm. This strategy would continue until one of the instances wins.
Remarks
You can also check out my favourite site for simulations of the Prisoner’s Dilemma.