Stealing cookies on Christmas Eve: An Introduction to Game Theory & The Nash Equilibrium

  • If the both of you stay silent, both of you will be denied treats for about a week for staying up past your bedtimes.
  • If both of you confess, both of you will be denied treats for 6 weeks.
  • If you confess and your sister doesn’t, you will be denied treats for 10 weeks while your sister receives no punishment
  • If your sister confesses and you don’t, your sister will be denied treats for 10 weeks, while you receive no punishment.

What is Game Theory?

  • they follow a set of rules and assumptions
  • the decisions of the players are interdependent — i.e. the choices of one player will affect the outcome of another.
  1. Cooperative vs. Non Cooperative: In cooperative games, players are permitted to negotiate and collaborate where as in non cooperative games, they are not.
  2. Normal vs. Extensive: In normal form games, you can play any of your moves all the time. In extensive form games, the moves you have available depends on those of your opponent.
  3. Sequential vs. Simultaneous: In sequential games, players move one after the other, where as in simultaneous games, they move at the same time.
  4. Zero sum vs. non zero sum: In zero sum games, the outcomes achieved by all the different players will sum to zero, whereas in non zero sum games, they do not.
  5. Symmetric vs. asymmetric: In symmetric games, the best strategy for each player is the same, whereas in asymmetric games, such strategies will not necessarily be the same.
  1. Non cooperative: you are not aloud to talk to your sister.
  2. Normal: you can always play any of your moves.
  3. Simultaneous: you have no knowledge of what your sister does before making your own decision — you are effectively moving at the same time.
  4. Non zero sum: your outcomes don’t sum to zero.
  5. Symmetric: as we will come to see in a moment, the best thing both of you can do is the same.
  • Players are those involved in game play (you and your sister)
  • Actions are decisions that the players make (staying silent or confessing)
  • Strategies are composed of actions. “Pure strategies” constitute playing one single action repeatedly and “mixed strategies” mean playing a bunch of different actions with specific probabilities (unless we try to steal cookies each Christmas eve, we only have the pure strategies of cooperating or confessing)
  • Payoffs are the outcomes which players receive as a result of their decisions and those of their opponents (being denied treats only one week vs. being denied treats for 10 weeks). Payoff tables represent all your possible outcomes relative to your possible actions.

Rationality and the Nash Equilibrium

  • If you switch strategies and don’t confess, but your sister confesses — you go from 6 weeks without treats to 10 weeks without treats.
  • If your sister switches strategies and doesn’t confess, but you confess — your sister goes from 6 weeks without treats to 10 weeks without treats.

How to find the Nash Equilibrium?

  • Rock is (0)(x) + 1(y)-1(1-x-y) = x + 2y -11
  • Paper is (-1)(x) + (0)(y) + 1(1-x-y) = 1–2x-y
  • Scissors (1)(x)-1(y) + (0)(1-x-y) = x-y

Applications of Game Theory

  • what is a military’s optimal missile strategy?
  • how can one most effectively bid at an auction?
  • how can an advertising firm maximize their sales?
  • what is a politician’s best possible campaign strategy?
  • what is a product’s competitive sales price?
  • how can an insurance company price their deals?
  • what is the optimal placement of telecommunication towers?
  • how could one negotiate labour management?

Key Takeaways

  1. Game theory studies “games” — competitive interactions between rational decision makers which follow a set of rules and has actions which are interdependent.
  2. A rational player will play the Nash Equilibrium — the optimal outcome of a game from which no player can benefit by changing strategies if none of his opponents do so as well.
  3. There is not a universally defined way of finding the Nash Equilibrium for a game — but there are techniques which are effective in certain situations such as minimax analysis and linear programming.
  4. Game theory has a many varied applications — ranging from the pricing a commodity to navigating war.

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Activator at The Knowledge Society | A Sandwich or Two Founder

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Avery Parkinson

Avery Parkinson

Activator at The Knowledge Society | A Sandwich or Two Founder

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