Oligopolies

What do profits depend on for oligopolists?

Depend on the quantity produced by the oligopolists and by other oligopolists. This is because price is the inverse of market demand, and so depends on total quantity (both by itself and others) and total cost just depends on quantity produced by itself.

Do competitors affect profits in perfect competition?

No, this is because no one has market power. They are all price-takers.

Why can oligopolists affect price?

Because they have market power

Why do oligopolists have market power?

Because its output is a large enough share of the market that changing it noticeably changes total supply. It changes quantity by enough so that price actually has a noticeable change.

Do oligopolists control price completely?

No, it also depends on what the other oligopolists produce. A firm chooses its own quantity, and market demand determines the resulting price from everyone’s quantities together.

Game theory

Makes predictions in those situations where gains/losses of agents depend on individual choice and choices of other individuals

What do we use game theory in economics for?

To make predictions when the utility of an agent spends also on choices of other agents. To amen predictions when profits of firms depend also on choices of other firms.

Strategic thinking

Agents (firms) think about what they expect other agents will do before they make their decision.

Static games

Agents make their decisions simultaneously (before taking action, you do not see the actions of the other players)

Dynamic games

Players play one at a time (the second player can see the action taken by their opponent before making a decision).

Rules of a static game

Are common knowledge. There is a set of players and a set of available actions for every player (finite or infinite). There are also payoffs (psychological aspects, monetary, points) for every player and for every possible outcome of the game.

What condition is there for games to be represented using a matrix?

That there are a finite set of actions - Normal form of the game = in matrix

Normal form of the game what do rows and columns represents

Rows = set of actions of player 1

Columns = set of actions of player 2

What does each box of the matrix correspond to?

A possible outcome.

Different outcomes have different payoffs which we can represent with

Utilities for each player

How do we avoid infinite strategic reasoning?

By using: solutions in dominant actions, solutions with dominated actions, Nash equilibrium

What is the best response?

An action is a best response to a given opponent’s action if it is the action that provides the agent with the highest possible payoff.

What is a dominant action?

An action is dominant if it is the only best response to every opponent’s action.

If both players have a dominant action, then…

they will choose that action.

If one player has a dominant action and the other player does not, then…

The player with a dominant action chooses that dominant action, the other player will best respond to the predicted action taken by the player with dominant action (aka choose the one with the highest payoff).

When is an action dominated?

When it is never a best response to an opponent’s actions. It cannot be justified as it never maximises the player’s utility.

First thing to do to find a solution in a matrix

Delete all dominated actions (rational player never chooses it)

What do you do after deleting dominated actions from the matrix?

Delete any “newly” dominated actions and repeat (either for one player or another)

Nash equilibrium

A pair of actions, one for each player, such that each action is a best response to the action played by the opponent.

Requirements of Nash Equilibrium

1. Every agent in equilibrium correctly predicts the behaviour of the other agent

2. Every agent will choose the action that maximises payoff, given hiss correct prediction about his opponent (principle of rationality)

Is the Nash equilibrium stable or unstable?

Stable, when agents reach the NE, they have no incentive to change their action (No unilateral deviations)

In Nash equilibrium does a player sometimes change their mind after seeing what the other player did?

No, the whole point is that it is stable. You were already maximising your payoff given a CORRECT prediction, therefore you would not change you action.

How to find the NE?

1. Find all the best responses to every opponent’s action and for every player.

2. Select boxes of the matrix where both actions are best responses.

How many Nash equilibria can a game have?

Infinite, but the less, the better as we can make better predictions.

What is the prisoner’s dilemma?

The prisoner's dilemma is a famous game theory thought experiment. It shows why two rational individuals might not cooperate, even when it is in their best interest to do so.

In the prisoner’s dilemma, why is the option where neither of them betray the other not a NE?

Because if agent a knew that agent b was not betraying him, then the option that would give agent a the most payoff is by betraying agent b. This would make the decision unstable as they would unilaterally deviate from their decision.

What is cooperation in the prisoner’s dilemma?

If both cooperate, neither betrays the other. They both get a better outcome than if both betray, but each has an individual incentive to betray when the other cooperates.

What is unilateral deviation?

A unilateral deviation is when one player changes their choice while the other player keeps theirs unchanged.

Why does cooperation not happen in the prisoner’s dilemma?

Because each prisoner has to decide without knowing what the other will choose, and betraying gives each one a better personal outcome in either case: If the other stays silent, betraying lets you benefit while they take the worse outcome. If the other betrays, betraying protects you from the worst outcome of staying silent alone.

For what situations do we use the prisoner’s dilemma?

Those where agents would be better off if they both cooperated, but incentive to deviate is stronger, which gives the NE (deviate, deviate).

Why can the prisoner’s dilemma be used to represent oligopolies?

Because each firm’s profit depends on what the other firms do. If both keep the price high, both earn higher profits. If one keeps the price high and the other low, the high priced firm loses customers to the lower priced firm. If both lower the prices, both earn lower profits.

How does the prisoner’s dilemma and cooperation manifest itself in oligopolies.

Cooperating would benefit them collectively, but each has an incentive to deviate for its own benefit (prisoner’s dilemma).

What is price fixing and why is it illegal?

If competing firms agree to keep prices high, that is price fixing, a form of cartel conduct prohibited under EU competition law. If they agree to keep prices high: collusion, generally illegal. If each independently chooses a high price: the high prices alone do not prove an illegal agreement.

Punishment strategy - Prisoner’s dilemma Oligopolies

“I’ll keep the high-price outcome as long as you do. If you cut your price once, I’ll switch to the low-price outcome forever.” That can discourage deviation because a firm weighs the extra profit from undercutting today against the lower profits it expects in future periods. It works only if that future loss matters enough to outweigh the immediate gain.

How does repeated interaction (playing the prisoner dilemma multiple time) allow for cooperation?

Repeated interaction can sustain cooperation because each player may give up the short-term gain from deviating to avoid losing the benefits of cooperation in future rounds.

Coordination game

Agents want to coordinate on a final outcome, but they have different preferences on which outcome. Every time we have a coordination game, we have multiple Nash equilibrium. This means we cannot predict the result correctly.

What do players want to do in a coordination game?

Do something together. Both \((A,A)\) and \((B,B)\) are Nash equilibria, because neither player can improve by deviating unilaterally from either outcome.

Zero-Sum Games

The sum of pay-offs in every box of the matrix is zero. If one player gains, the other loses by the same amount.

Does changing decision improve pay-off in a Nash equilibrium?

A Nash equilibrium is a situation where neither player can do better by changing their choice alone.

Is there a pure-strategy NE in zero-sum games?

No, however, a finite game still has at least one NE when mixed strategies are allowed.

Zero Sum Game mixed strategy

A mixed strategy means choosing actions randomly according to probabilities. You assign probabilities per choice, a mixed strategy tells us how likely a player is to choose each action. The probability of the outcomes are the product of the probability of each choice of each player that leads to said outcome.

What are pure actions?

You pick 1 action with certainty

Dominant actions and NE

A dominant action is always part of a Nash equilibrium (dominated action = never)

With iterative deletion of dominated actions you never delete an NE/

If you find a unique solution in dominate actions or with iterative deletion → NE.

How to find NE if you do not have one unique solution after iterative deletion of dominated actions?

For each remaining outcome, hold one player’s choice fixed and ask: Could the other player get a higher payoff by switching alone? Do this for both players. The outcome is a Nash equilibrium only if the answer is no for both.

When is NE good?

When a game is repeated often and agents learn how to play and how their opponents play. We see convergence to NE>

Unqiue self-enforcing agreement.

What is a unique self-enforcing agreement?

A self-enforcing agreement is one that each player wants to follow because following it is best for them given that the others follow it. No outside enforcer is needed.

Unique self-enforcing agreement will always be followed

If players can coordinate in advance but cannot enforce their agreement, the unique Nash equilibrium is the only outcome they can both be sure the other will follow: once the game begins, neither has an incentive to deviate alone.

Weakly dominated choice

AN action is weakly dominated if it is dominated in one case but indifferent in another case. An action is weakly dominated if it gives the agent a lower payoff than other actions, except in some cases where it gives the same payoff as other actions

Weakly dominant action

An action is weakly dominant if it is dominant in one case but indifferent in another case. It is weakly dominant if it always gives the agent higher or the same payoff as other actions.

Do weakly dominated factions get deleted in iterative deletion?

No, they survive the iterative deletion of dominated actions and can be part of a Nash equilibrium. This is because, against the other player’s equilibrium choice, the two actions may tie. Switching then gives no improvement, so there is no profitable unilateral deviation.

Can strictly dominated actions be part of a Nash equilibrium?

No, never

When is an outcome efficient?

If it is not possible to make someone better off without making someone else worse off.

Static games with an infinite set of actions a_i [0, epsilon]

the other player’s actions a_-i are denoted by a_-1 _ in [underline a , overlie a] so lowest and largest allowed actions

Every player has a best response found by a best response function. BR(a_-i) that associates to every action by the other play the best response by player 1. The highest

Take the BR_2 (action 1) to find a2* and do BR_1 (action 2 ) to find a1*

best response graphs

br2(a_1) and br1(a_2), so best response to all actions, and find the point of intersection