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Game Theory

The Exchange Game

__Scenario:__

• **Two players and two envelopes**

• Both players know that envelopes may contain** 5, 10, 20, 40, 80, or 160 Euro**

• Both player know that **one envelope contains twice the amount of the other envelope**

• The two envelopes are r**andomly distributed** to the two players

• After both players have inspected their envelopes they are asked whether they want to exchange

• Making an **exchange offer costs 1**

• If both agree to exchange, they exchange

__Supposition:__

• Suppose one player has 10 and the other 20

• Expected gain from exchange for player with 20 is **½·40 + ½·10 – 1= 24 > 20**

• Expected gain from exchange for player with 10 is **½·20 + ½·5 – 1= 11.5 > 10**

• Since both players profit from an exchange, they may both agree to exchange → **exchange takes place**

• How can it be that both agree, since, a priori it is clear that exactly one player wins from the exchange, while the other one loses?

• __Problem:__ players did not think about the strategic situation. Since both parties involved act strategically, the problem cannot be solved by shortsighted expected values, but by a strategic analysis

__Strategic Analysis:__

consider the border cases first. Suppose player finds 160; knows for sure that other player has 80 → does not want to exchange

⇒ player with 80 knows that he can only exchange with a player with 40→ does not want to exchange

⇒ player with 40 knows that he can only exchange with a player with 20→ does not want to exchange

…

⇒ player with 5 would like to exchange with 10, but knows that player with 10 does not make an offer → player with 5 saves the cost of 1 and also does not make an offer to exchange

-> Strategic analysis shows that there are **no incentives for exchange**, although each player ≠ 160 has a **positive expected value from the exchange**

Game Theory

8.3 Voting Schemes

• There are many ways groups can apply to choose between alternatives.

• We will introduce 5 prominent voting schemes:

– Simple majority voting

– Plurality runoff

– Sequential runoff

– Borda count

– Condorcet procedure

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

Game without equilibrium (3)

**• “Highest number wins”:**

– Two players write down a number.

– The player with the higher number wins.

• This is an **infinite game** in which each player has infinitely many actions.

• This game has** no equilibrium** (neither in pure nor in mixed strategies).

Game Theory

Indifference

• According to the fundamental lemma, **mixed strategy equilibria consist of pure strategies that are all best replies to the (mixed) strategies of the others**. This means that **all pure strategies lead to the same payoff as the mixed strategy, given that the others do not deviate**.

• Why should a player then play the mixed strategy and not just one of the pure strategies that are best replies?

-> The answer is that otherwise, the **other players would have an incentive to deviate** and the whole system would become **unstable**: players have to mix, such that the prob. of “scoring right” equals the prob. of “scoring left”.

Game Theory

How to understand mixed strategy

equilibria (4)

• Randomization (e.g. matching pennies)

• Uncertainty about the action choices of the opponent(s)

• Probabilities as historic frequencies

• Probabilities may represent the proportion in which certain (pure strategy playing) types are present in the opponent population

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

__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

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.

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