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Game Theory

__Normal Form Games (4)__

• A normal form game is a game in which the players decide **simultaneously** and independent of each other.

• This means that **no player knows the decision of the other players** at the time of decision.

• The description of a strategic situation by a normal form game is suited if the players

– **decide at the same time**

– or **decide in sequence without knowing the decisions of the predecessors**

• The fact that each player has a utility over the outcomes (i.e. the actions of __all __players) and not only on the payoff of the own action, distinguishes a strategic situation from a pure decision situation.

Game Theory

How to determine mixed strategy

equilibria?

• According to the previous lemma **all actions that are chosen with positive probability in a mixed strategy lead to the same utility.**

• This is the key to the calculation of the mixed strategy equilibria, i.e. **the determination of the actions that are chosen with positive probability and the corresponding weights.**

Game Theory

Group Choice

• In the non-cooperative game theory analysis the decider is a**lways an individual** with preferences over the possible outcomes of the game.

• Even if we model a firm as a player, we assume that it has a single utility function.

• In fact, the decisions of a firm are the result of a process in the firm (its board of directors). How this process works is neglected.

• In cooperative game theory we studied coalition formation and group decisions for allocating payoffs.

• In this chapter we study **mechanisms that aid a group** (e.g. a committee, a parliament, a board, …) to decide among alternatives (e.g. candidates in an election, new laws, different business plans).

Game Theory

Shapley Value - Intuition

• Suppose the N players wait in front of a room.

• In a random order these players enter the room; one after the other.

• For each player that enters calculate the amount by which this player increases the coalition value of the players already in. This is the player’s marginal contribution.

• The idea of the Shapley value is to allocate the marginal contribution to each player.

• If one would give each player exactly her marginal contribution, the allocation would depend on the random order in which the players enter.

• Hence, the Shapley value allocates the average marginal contribution (averaged over all N! sequences in which the players can possibly enter the room) to each player.

Game Theory

6.3 The Shapley Value

• The core and the bargaining set defined sets of stable coalitions with associated payoff allocations.

• The Shapley value is a uniquely defined **payoff vector for each characteristic function game**.

• **The sum of the players’ payoffs in the Shapley value always adds to v(N).**

• The **marginal contribution** of player i is the value that is added to the coalition value if player i joins the coalition C.

Game Theory

Core and Bargaining Set

• The bargaining set assumes that the argument underlying **an objection for which there is no counter-objection undermines the stability of an outcome**.

• The bargaining set is a **weaker concept than the core** since it allows objections, but only those with counter-objections.

• The core does not allow any objections.

•** Hence the core is a subset of the bargaining set.**

Game Theory

Cooperative Game Theory

• In the __non-cooperative game theory __models the players decide autonomously to their best interest and without coordinating with others.

• The rules of the game are completely

specified.

• Small changes in the rules of the game may lead to dramatic changes in the equilibrium prediction.

• The main focus of __cooperative game theory__ are groups of players and the **strategic possibilities of the group**.

• The rules of the game may be vague.

• An outcome of a cooperative game (or coalitional game) is a coalition of players and the joint actions of this coalition.

• The preferences of the players refer to the joint actions of the group.

• Non-cooperative game theory analyzes **stability issues in the individual strategy choices**, whereas cooperative game theory analyzes the **stability of coalitions.**

Game Theory

Folk Theorem (general)

• The folk theorem shows that there is a **huge multiplicity of equilibria in infinitely repeated games**.

• The threat of using a** trigger strategy is often too strong** since it suffices to punish so long that a **deviation becomes unprofitable**. Hence the punishment can stop after a finite time.

• The threat with a trigger strategy is often incredible.

Game Theory

Minimax payoff

– each player calculates for each combination of pure strategies of the other players **his best response** (highest payoff).

– The minimax-payoff is the **lowest of these best response payoffs**.

– This is the payoff a player can **guarantee himself against known strategy combinations of the other players**.

– The strategy combination that leads to a players minimax-payoff is called **maximal punishment.**

Game Theory

Maximin payoff

– each player calculates **for each combination of pure strategies of the other players** **his lowest payoff.**

– The maximin-payoff is the** highest of these lowest payoffs**.

– This is the payoff a player can **guarantee himself**.

– No matter what the other players do, a **player can always achieve at least his maximin-payoff**.

Game Theory

Repeated Games (general)

• The repeated game is called **supergame**.

• The game that is repeated is called **stage game**.

-> The stage game may be either a **normal form or an extensive form game**

• At the** end of each stage game**, the **players are informed about the other players’ actions** in the stage game.

• Hence a repeated game is an **extensive form game with complete information in which the same game is played on each stage.**

Game Theory

2 forms of Repeated Games

1. **finite games**, i.e. games with an ex-ante known number of repetitions

2. **infinite games**, i.e. games for which the players belief after each repetition that the game continues.

-> The number of repetitions T is known and can be sufficiently large.

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