Game Theory I Flashcards
(33 cards)
Definition of a game
A competitive activity in which players contend with each other according to a set of rules
Definition of game theory
A formal representation of a situation in which a number of individuals interact in a setting of strategic interdependence
What is game theory? (+ general facts)
- Formal study of strategic interactions
- It is the study of how we mathematically determine the best strategy given conditions to optimize the outcome
- It helps to analyze situations more rational and formulate acceptable alternatives
- Everything can be considered a game (in a game theoretical sense)
- At least twelve game theorists have won the Nobel prize in economics so far
History of game theory
- Game theoretical notions go back hundreds of years
- Modern game theory credited to John von Neumann
- John von Neumann and Oskar Morgenstern
- John Nash generalized these results and provided the basis of the modern field of game theory
What characteristics are required in order for something to be considered a game?
- A set of players {A, B, C, …}
- A set of strategies
- The payoff (ie. what a player gets) is listed for each possible list of strategies of each player
– Utility
In what ways can games vary?
- Simultaneous vs Sequential games
– Players move at the same time or the action of one depends on another - Single play vs Iterated games
– Game is played only once (static) vs multi-stage games - Zero vs Positive sum games
– The sum of the payoffs remains constant during the game versus they change - Complete vs Incomplete information games
– Players know or do not know each other’s payoffs - Non-cooperative vs Cooperative games
– Players can or cannot form binding agreements and coalitions
What is the play of a game?
A play of the game is a pair, such as (Up, Right) where the first element is the strategy chosen by player A and the second is the strategy chosen by player B
What is another term for a likely play?
Nash equilibrium
- Nash equilibrium is a solution concept made by John Nash around 1949-1950
John Forbes Nash facts
- Born in 1928 in Virginia
- Engineer at heart, chemistry, but ultimately mathematics
- His PhD thesis (Princeton) formulated the concept of Nash equilibrium
- After defense, he produced several papers setting out several different theories
- In 1957: he married
- In 1959: acute paranoid schizophrenia
- From 1990 onward: progressive healing
- 1994: Nobel Prize in Economics
- 2001: A Beautiful Mind
- 2015: Abel prize and died in a car crash
How are Nash’s achievements perceived?
- Game theory: 95% of Nash’s fame
- But some (including himself) have argued that this is one of his least impressive accomplishments
- He has made some very big strides in analytics and geometry
Definition of a Nash equilibrium
A Nash equilibrium is a set of strategies one for each player, such that no player has incentives to change his or her strategy
- It is inherently stable
What is a Pure Strategy Nash Equilibrium? (PSNE)
Players only play a single strategy in equilibrium
What is a Mixed Strategy Nash Equilibrium? (MSNE)
Players play a combination of several strategies with a fixed probability, ie. they randomize between strategies
What are the two problems with Nash equilibria?
- A game can have several Nash equilibria so what is the outcome then?
– Several additional solution concepts:
—> Focal point strategies
—> Correlated equilibria
—> Eliminating dominated strategies - There may not be a Nash equilibrium in pure strategies
Story of the Prisoner’s Dilemma
- Two people {A,B} are arrested for a crime
- There is not enough evidence to convict either of them, they can only sentence them for minor misdemeanors
- The police needs to get them to confess their crimes to be able to charge them both
- How do the police get the suspects to confess
- They are put in different cells and they cannot communicate with each other
What deal does a police officer offer in the Prisoner’s Dilemma?
- If no one confesses (silence) for the robbery, the policy can only charge the prisoners for trespassing
– Low sentence ~1 month - If one confesses and the other does not, the police will be lenient on the rat and severly punish the quiet one
– eg. 12 months in jail for the quiet one, 0 months for the rat - If both confesses, the police will punish both equally
– eg. 8 months in fail each
What is a dominant strategy?
Strategy X strictly dominates strategy Y for a player if X generates a greater payoff than Y regardless of what the other players do
- Dominant strategies are considered as better than other strategies, no matter what other players might do
Nash equilibrium in the Prisoner’s Dilemma
- The only Nash equilibrium for this game is (confess, confess) even though (silent, silent) gives both players better payoffs
- Even if they make a deal before they go to the interrogation room; this deal will not uphold because they have individual incentives to deviate from keeping silent
- The only Nash equilibrium is so-called Pareto inefficient
Example of Prisoner’s Dilemma (Trump + Republicans)
- George Conway: Conservative lawyer (former republican)
- Argues in a podcast called “Trump’s 91 problems (& jail is one)” that:
– “This has been the prisoner’s dilemma problem of the Republican party. Nobody actually wants to go out and tell the truth about this guy because they don’t want to be out there standing alone” - Even though Republican senators would be better off fighting Trump’s attack on democracy they aren’t doing it, they are trapped
What is the stag hunt game?
The Stag Hunt is a game that describes a conflict between safety and social cooperation
- Some say that it better represents reality in comparison to PD games
What are other terms for the Stag Hunt game?
Also known as the “assurance game”, “coordination game”, and “trust dilemma”
What is the principle of the Stag Hunt game?
If people’s actions complement each other, then getting a good outcome depends not just on what people value but on how they expect each other to behave
Background to the stag hunt game
- A group of hunters are on the hunt for stags
- If they work together, they can kill the stag, if not, the stag flees, and they will go hungry
- The stag is not easily found, while waiting they see a hare
What is the game of the stag hunt?
- If the hunter leaps out and kills the hare, he will eat, but the trap laid for the stag will be wasted and the other hunters will starve
- There is no certainty that the stag will arrive; the hare is present
