Economics Ch.14: The Theory of Games

The prisoner's dilemma is all around us, and game theory can help you navigate the complex world of competition.

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You CAN have 2 Nash Equilibriums in in one payoff matrix!!

true, make sure to go through each scenario, don't stop when you find one Nash Equilibrium.

3. Identify all Pareto optimal outcomes.

(High, Left) is not Pareto optimal because both players are made better off by switching to (Low, Left). (High, Right) is Pareto optimal because each other outcome makes Player B worse off. (Low, Left) is Pareto optimal because each other outcome makes Player A worse off. (Low, Right) is not Pareto optimal because both players are made better off by switching to (Low, Left).

2. Identify all Nash equilibria.

(High, Left) is not a Nash equilibrium because both players will regret their choice. Player A will wish they had played Low, and Player B will wish they had played Right. (High, Right) is not a Nash equilibrium because Player A regrets their choice. (Low, Left) is a Nash equilibrium. Player A is happy to have gotten 6 instead of 5, and Player B is happy to have gotten 5 instead of 3. (Low, Right) is not a Nash equilibrium. Player B regrets their choice.

Rules for Success in Cooperative Games (Game Theory)

-Be nice: cooperate, never be the first to defect -Be Provocable: return defection for defection, cooperation for cooperation -Don't be envious: Focus on maximizing your own 'score' as opposed to ensuring that your score is higher than your 'partners' -Don't be too clever: don't try to be tricky. Clarity is essential for others to cooperate with you (earn your trust).

What are the most profitable auction strategies for the auctioneer?

-Dutch Auction -Sealed Bid (1st price)

What are the most profitable auction strategies for the bidder?

-English Auction -Sealed Bid (2nd Price)

What are the best auction choices for Tony (where he will bet the least comfortably)?

-English Auction OR -Sealed Bid (2nd Price)

John Nash

-mathematician; one of the first to outline the fundamentals of game theory. -Nobel prize in economics in 1994

Payoff Matrix

-shows the payoff for each player in each possible outcome -Shows us every possible outcome in the Prisoner's Dilemma.

How do Game Theorists break down a game?

A game consists of players, strategies, and a payoff matrix -visualizing a game in this way helps us to see what a rational-decision maker will do.

Zero-Sum Game

A game where every outcome has payoffs for each player which add to zero -there's no way to make one player better off without making another player better off -all of the outcomes are Pareto Optimal

Dutch Auction

An open outcry auction in which the price decreases until a bidder stops the auction. The bidder winds the item and pays the bid

English Auctions

An open outcry auction in which the price increases until there is only one standing bid. That bidder wins the item and pays the bid.

Pareto Optimal

An outcome is Pareto Optimal when you cannot make one player better off without making another player worse off. -an outcome/situation when there is no way to make another player better off without making another player worse off

Nash Equilibrium

An outcome where no player regrets their choices given the choices of other players. -reached when each player follows their dominant strategy -the Nash equilibrium must be found within the payoffs for that dominant strategy -look at player's decisions and ask "would they regret their choice?"-if one or both of them would regret their decision, it is NOT the Nash Equilibium -the outcome when both players follow their self-interest

Game Theory

Application of mathematical models to strategic interaction among rational decision-makers in an attempt to analyze and predict behavior

Sealed Bid (1st Price)

Bidders privately submit bids at the same time. The highest bidder wins the item and pays the amount of their bid.

Sealed bid (2nd Price)

Bidders privately submit their bids at the same time. The highest bidder wins the item but only pays the amount of the second highest bid.

The Prisoner's Dilemma

Experiment that challenges two rational agents to a dilemma: they can operate with their partner for mutual benefit OR betray their partner for individual reward

Classic Example of the Prisoner's Dilemma

If Prisoner A decides to Confess, but Prisoner B cooperates, then Prisoner A will get special treatment from the prosecutors. Prisoner A will serve 0 years in prison, while Prisoner B is sentenced to 5 years.

Dominant Strategy

One thing we want to look for in any game is a dominant strategy. -A dominant strategy is a strategy that is better for a player no matter what strategy their opponent chooses. EXAMPLE: player B's dominant strategy is to confess because that way they will either get 0 or 3 years in prison which is better than 1 or 5 years.

Revenue Equivalence Theorem

Under certain assumptions, the four auction types are expected to raise the same revenue for the winner.

1. Identify where or not any player has a dominant strategy.

What will Player A want to do if Player B chooses Left? Well, if Player A chooses High, they will get a payout of 5, but if they choose Low they will get a payout of 6. So, if Player B chooses Left, then Player A wants to choose Low. What if Player B chooses Right? Player A will still want to play Low, because the payout of 4 is better than the payout of 3. Since Player A always wants to choose Low, it means Low is Player A's dominant strategy. We can apply the same process to Player B. When Player A chooses High, Player B will want to have chosen Right, since 6 is better than 4. When Player A chooses Low, Player B will want to have chosen Left, since 5 is better than 3. Since Player B's best choice depends on what Player A chooses, Player B does not have a dominant strategy.

How do you find the Nash Equilibrium in the payoff matrix?

You go through each outcome and find the one that neither player would regret. -the outcome where both players use their dominant strategy to

Robert Axelrod

a political scientist -The Evolution of Cooperation: why humans would evolve to cooperate with one another at all? -Prisoner's Dilemma Tournament

Mutually Assured Destruction

an attempt to set the payoff matrix in a way where the dominant strategy is to not use nuclear weapons because the payoffs are so bad for just using one.

Sequential Games

games played in sequence, where one player goes first and then the next player chooses based on that move.

Decision Node

points in a decision tree when decisions need to be made; used for sequential games

What is meant by games are "non-cooperative"?

the players are not able to coordinate with each other to try and agree on certain outcome. Each player has to make their decision in isolation of the other players.

Game theory has a lot of power when it comes to explaining the world around us. Biologists have applied it maybe more than economists because of how well it explains which traits will be favored by natural selection.

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Often people have a sense of what other people are willing to pay.

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Rational decision-makers attempt to optimize their own outcome, and so each prisoner will seek the best possible payoff for themselves.

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Taken at face value, the prisoner's dilemma explains the logic of a classic interrogation technique used by the police to try and get people to confess to crimes, and if these prisoners really are guilty than probably getting both to confess is what is best for society. But as a thought experiment, it blasts a hole into economic theory, showing us instances where a lack of cooperation can lead to sub-optimal outcomes. And that is something we are going to want to explore.

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