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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

__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

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

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

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

Backward Induction (3)

• Backward induction is an algorithm to **determine subgame perfect equilibria.**

• The game tree is analyzed from backward and the equilibria of the subgames are determined.

• This is also the way to proof Zermelo’s theorem.

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

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

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

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

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

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