PHIL384 - Game Theory

A game contains PSGO

Players, Strategies, Games, Outcomes

Strategy

A comprehensive plan of action

Dominant strategy

A strategy that yields the best outcome, regardless of what the other player chooses

The most dominant strategy in a Prisoner’s Dilemma one-shot

Defecting

Zero-sum game

a situation in which one participant's gain is precisely matched by another participant's loss, resulting in a total net change of zero for the system; there is a winner and a loser

Strictly dominant strategy

ALWAYS has a higher payoff than other strategies

Weakly dominant strategy

At least sometimes has a higher payoff—and never lower—than other strategies

Pure vs Mixed Strategies

Mixed strategies have some degree of randomness in them while pure strategies have no randomness

Mixed strategy example

JOSS

Pure Strategy Example

TIT FOR TAT

Perfect vs Imperfect Information

A game has perfect information if a player can see the opponent’s previous moves, a game has imperfect information if a player cannot see all of the previous moves

A Game of Complete Information

Chess or checkers

Complete vs Incomplete Information

A game has complete information if the player knows the opponent’s strategies and payoffs, a game has imperfect information if a player does not know the opponent’s strategies or payoffs

Nash Equilibrium

A strategy profile where each player is playing their best response to the others’ strategies and in turn getting the best payoff in that given scenario

The Nash Equilibrium in the Prisoner’s DIlemma

1/1; both players are playing their best response, defection

Pareto Improvement

An Outcome (O) is better for someone but worse for no one

Pareto efficient

No pareto improvements are possible, no one else can benefit without someone else being worse off

Strict pareto

Both improve

Pareto inferior

An outcome worse for someone

The top eight strategies in Axelrod’s tournament were…

Nice

Discount parameter

A reflection of how much players care about the future in regard to the present

WIth a higher discount parameter…

The future matters just as much as the present to the player

Simultaneous vs Sequential Games

In simultanous games, the player does not have time to react, while in sequential games the players play one after the other

Example of a Simultanous Game

Rock paper scissors

Example of a Sequential Game

Chess or checkers

Non-Zerosum Game

a situation where the total gains and losses among all participants do not necessarily add up to zero, meaning a player's win is not always another's loss; there is not always a winner and a loser

One-Shot Game

an interaction that occurs only once, meaning players cannot learn from past interactions or influence future ones through repetition

Repeated Game

a scenario where the same game is played multiple times; players remember past actions and outcomes, allowing them to develop more complex strategies, which in turn influence future decisions and can lead to different results than in a one-shot game

Dominant Strategy vs Best Response

a dominant strategy is a best response to all possible opponent strategies, whereas a best response is a best response to a specific opponent strategy

What does the Prisoner’s Dilemma have in common with other games?

The tension between self-interest and collective interest, making decisions to benefit oneself or everyone; a payoff system that outlines the rewards or punishments of particular choices

The Prisoner’s Dilemma is so special because…

it illustrates the conflict between individual self-interest and the collective good, showing that a rational pursuit of self-interest can lead to a worse outcome for everyone involved than mutual cooperation would have produced

Who won in Axelrod’s tournaments and in virtue of what?

TIT FOR TAT won due to its niceness, forgiveness, retaliation, and clarity in execution

How did Axelrod’s Tournaments work?

Axelrod create a series of computer tournaments with repeated games where different strategies for the Prisoner’s Dilemma competed against each other

TIT FOR TAT

Cooperates initially and then mirrors the other player’s previous move

Why did Axelrod need to run a tournament?

to discover which strategies were most effective in a partially cooperative and competitive environment; traditional game theory had concluded that mutual defection was the only rational outcome in a one-off game, but this didn't reflect the emergence of cooperation in repeated, real-world interactions; to find the most robust strategy

Collective Stability

A situation where no single player has an incentive to deviate from a given strategy if everyone else continues using it

Invasion

the ability of a new strategy to spread within a population dominated by another strategy; a strategy invades when a small number of players using that strategy can achieve higher payoffs than the existing population, allowing them to increase in relative frequency over time

Why the “evolution of cooperation?”

Axelrod is looking at how cooperative behavior can emerge and persist over time, even among self-interested individuals, through processes analogous to natural selection

When a strategy is unclear…

It is untrustworthy and difficult to understand, making every turn like the last time they will see each other

A strategy is the best strategy dependent on…

What other strategies are playing

The top eight strategies in Axelrod Tournaments were….

Nice

The evolution of cooperation is…

Like a ratchet, asymmetric in one direction

A random strategy…

Cooperates 50% of the time, defects 50% of the time

The best response against ALL C

ALL D

Best Response

The strategy that gets you your best score relative to your opponent’s strategy

In a one-shot prisoner’s dilemma, you’re best off…

Defecting no matter what

Unconditional Strategy

Does not change no matter what

Conditional strategy

Changes dependent on characteristics of the game

The best response to a random strategy

ALL D

Axelrod’s Tournament was a…

Round Robin Tournament; there was no elimination and the best final combined score over all games won

DOWNING

Starts with a double defect and defects forever on unresponsive players

A nice strategy is…

Not the first to defect

TIT FOR TAT

starts off cooperating, then copies opponent’s prior move, has a one-turn memory

JOSS

A variation of TIT FOR TAT; a mixed strategy; defects more than TIT FOR TAT

If JOSS and TIT FOR TAT play against each other…

they will go back and forth defecting and cooperating with each other

TIT FOR TAT won because it…

was nice, retaliated when necessary, forgiving, and was clear

To invade you have to…

Do better against the native strategy than the native strategy does against itself

If there is a cluster of TIT FOR TATS…

they take over as their success with each other outweighs the initial loss against the mean strategies

In order for a cluster of TIT FOR TATS to successfully invade, there needs to be a threshold of…

5%, meaning 5% needs to be TIT FOR TATS

TIT FOR TAT is collectively stable, especially if there is a…

High discount parameter