346_lect20.pdf

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Lecture Notes 20: Cooperative Games

Cooperative and Noncooperative Games

Cooperative games are different from noncooperative games. In noncooperative games, we assume that each player always acts in his own best interest and therefore that he chooses strategies with the objective of maximizing his own payoffs. Noncooperative games can create issues like the prisoners’ dilemma, where each player acting unilaterally in his own best interest leads to an equilibrium where all players are worse off. There is no reason to expect equilibria in noncooperative games to be efficient, and often they are not.

Cooperative games are different. In cooperative games, players have the ability to form binding agreements to divide surplus among themselves. Thus, the outcome in cooperative games is always efficient. For example, the prisoner’s dilemma disappears because players can make a binding agreement with each other to play the cooperative outcome and divide the surplus from doing so. In cooperative games, the players work as a single unit to maximize their total payoff.

Which concept is more appropriate depends on the situation. For interactions where players do not have the opportunity to make deals with each other beforehand or where there is no institution for making these deals enforceable (e.g. a binding contract), noncooperative game theory is probably more relevant. But for interactions where players can negotiate with each other and where there is some ability for them to make binding agreements, cooperative game theory is probably more appropriate – the players will work together to reach the efficient outcome.

A good example is family interactions. Being in a joint household generates a household surplus (e.g. cost savings from economies of scale, risk-sharing and the ability to produce children). But how is the surplus divided? In cooperative game theory, the husband and wife would look at their outside options and craft an agreement that benefits both of them. The outcome will always be efficient. It seems like cooperative game theory is more appropriate here. But the question is whether spouses can form mutually beneficial agreements that are credible and binding. If not, and if each spouse has an opportunity to defect from the agreement for his own benefit, then whatever deal they make needs to be self-enforcing, meaning that noncooperative game theory is more relevant. This introduces the possibility of inefficiencies.

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Coalitions and the Coalitional Representation

A cooperative game involves coalitions. A coalition is some subset of the players. In a cooperative game with 𝑛𝑛 players, there are 2𝑛𝑛 − 1 coalitions. For example, consider a game with 𝑛𝑛 = 4 players, whom we call A, B, C and D. There are 24 − 1 = 15 coalitions. They are:

{𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷, 𝐴𝐴𝐵𝐵, 𝐴𝐴𝐶𝐶, 𝐴𝐴𝐷𝐷, 𝐵𝐵𝐶𝐶, 𝐵𝐵𝐷𝐷, 𝐶𝐶𝐷𝐷, 𝐴𝐴𝐵𝐵𝐶𝐶, 𝐴𝐴𝐵𝐵𝐷𝐷, 𝐴𝐴𝐶𝐶𝐷𝐷, 𝐵𝐵𝐶𝐶𝐷𝐷, 𝐴𝐴𝐵𝐵𝐶𝐶𝐷𝐷}

The last coalition 𝐴𝐴𝐵𝐵𝐶𝐶𝐷𝐷, which includes all of the players, is called the grand coalition.

The security level𝑉𝑉(𝐶𝐶) of some coalition 𝐶𝐶 is the surplus that this coalition could realize on its own. This is similar in concept to a BATNA, just extended to more than one player. For example, in a household with four roommates, two of them might be friends who can leave to go live on their own. This coalition’s security level is the surplus they could achieve on their own.

The coalitional representation of a game describes the surplus attainable by all possible coalitions in a cooperative game. For example, the following is the coalitional representation of a game that involves three players.

𝑉𝑉(1) = 0 𝑉𝑉(2) = 0 𝑉𝑉(3) = 0 𝑉𝑉(12) = 70 𝑉𝑉(13) = 60 𝑉𝑉(23) = 50 𝑉𝑉(123) = 100

No player can get any surplus on his own. Players 1 and 2 can get a surplus of 70 on their own. Players 1 and 3 can achieve a surplus of 60 on their own. Players 2 and 3 can get a surplus of 50 on their own. The three players together can achieve a surplus of 100.

Obviously, the efficient outcome is for all three to form a coalition. The question is how they will divide the surplus of 100 between them. An allocation{𝑥𝑥₁, 𝑥𝑥₂, … , 𝑥𝑥_𝑛𝑛} is a division of the surplus among the 𝑛𝑛 players. An allocation is feasible if the allocation is possible given the setup of the game. In this game, feasibility requires that the surplus allocated to the three players not exceed 100 in total. But, among all feasible allocations, what will be the allocation of the surplus that the three players agree on?

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The Core

An allocation {𝑥𝑥₁, 𝑥𝑥₂, … , 𝑥𝑥_𝑛𝑛} is blocked by some coalition 𝐶𝐶 if 𝑉𝑉(𝐶𝐶) > ∑ 𝑥𝑥_{𝑖𝑖∈𝐶𝐶} _𝑖𝑖.

The idea is that a coalition 𝐶𝐶 can achieve a surplus 𝑉𝑉(𝐶𝐶) on its own. The total surplus that the members of the coalition end up with in the final allocation is ∑ 𝑥𝑥_{𝑖𝑖∈𝐶𝐶} _𝑖𝑖. If the coalition can do better on its own by breaking away from the group than what it gets in a proposed allocation, then the coalition can block the allocation by threatening to pull out.

This idea is again just a generalization of the BATNA concept. In a bargaining game, each player’s final allocation had be at least as good as his BATNA, or else he is better off withdrawing completely. The core is this same idea, just generalized to groups of players (coalitions) blocking allocations, in addition to individual players.

The core is the set of allocations that cannot be blocked by any coalition. In other words, the division of the surplus has to provide an allocation to each coalition at least equal to what that coalition could have gotten on its own.

For the cooperative game given above, the following inequalities characterize the core.

𝑥𝑥1 ≥ 0 𝑥𝑥2 ≥ 0 𝑥𝑥3 ≥ 0 𝑥𝑥1+ 𝑥𝑥2 ≥ 70 𝑥𝑥1+ 𝑥𝑥3 ≥ 60 𝑥𝑥2+ 𝑥𝑥3 ≥ 50 𝑥𝑥1+ 𝑥𝑥2+ 𝑥𝑥3 ≥ 100

Feasibility requires that the total surplus allocation not exceed the maximum attainable surplus of 100.

𝑥𝑥1+ 𝑥𝑥2+ 𝑥𝑥3 ≤ 100

The combination of the last no-blocking inequality (for the grand coalition) together with the feasibility inequality give us that:

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Efficiency requires 𝑥𝑥₁ + 𝑥𝑥₂+ 𝑥𝑥₃ to be at least 100 since this is the surplus attainable by the grand coalition. But feasibility requires the sum 𝑥𝑥₁+ 𝑥𝑥₂+ 𝑥𝑥₃ to be at most 100 since this is the surplus actually available. Combining the two together implies that 𝑥𝑥₁+ 𝑥𝑥₂+ 𝑥𝑥₃ must be exactly equal to 100. Any core allocation fully realizes the available surplus.

Thus, the core is the set of allocations {𝑥𝑥₁, 𝑥𝑥₂, 𝑥𝑥₃} that satisfy the following:

𝑥𝑥1 ≥ 0 𝑥𝑥2 ≥ 0 𝑥𝑥3 ≥ 0 𝑥𝑥1+ 𝑥𝑥2 ≥ 70 𝑥𝑥1+ 𝑥𝑥3 ≥ 60 𝑥𝑥2+ 𝑥𝑥3 ≥ 50 𝑥𝑥1+ 𝑥𝑥2+ 𝑥𝑥3 = 100

To get a cleaner expression, we can take the inequality 𝑥𝑥₁+ 𝑥𝑥₂ ≥ 70 and substitute from the final equality 𝑥𝑥₂ = 100 − 𝑥𝑥₁− 𝑥𝑥₃:

𝑥𝑥1+ 𝑥𝑥2 ≥ 70 𝑥𝑥1+ (100 − 𝑥𝑥1− 𝑥𝑥3) ≥ 70 𝑥𝑥3 ≤ 30

But we know from the third inequality that we need 𝑥𝑥₃ ≥ 0. Combining these two expressions, any core allocation must have 𝑥𝑥₃ ∈ [0,30].

Doing the same substitution with the inequalities 𝑥𝑥₁+ 𝑥𝑥₃ ≥ 60 and 𝑥𝑥₂+ 𝑥𝑥₃ ≥ 50 will give that

𝑥𝑥2 ∈ [0,40] and 𝑥𝑥1 ∈ [0,50].

Thus, a simple characterization of this problem’s solution is that the core consists of allocations with 𝑥𝑥₁+ 𝑥𝑥₂ + 𝑥𝑥₃ = 100 such that 𝑥𝑥₁ ∈ [0,50], 𝑥𝑥₂ ∈ [0,40] and 𝑥𝑥₃ ∈ [0,30].

Example: Markets and Trading

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𝑉𝑉(𝐵𝐵) = 0 𝑉𝑉(𝑆𝑆) = 0 𝑉𝑉(𝐵𝐵𝑆𝑆) = 200

The core inequalities for the allocation of the surplus {𝑥𝑥_𝐵𝐵, 𝑥𝑥_𝑆𝑆} to the buyer and to the seller are:

𝑥𝑥𝐵𝐵 ≥ 0 𝑥𝑥𝑆𝑆 ≥ 0 𝑥𝑥𝐵𝐵+ 𝑥𝑥𝑆𝑆 ≥ 200

Combining the last inequality with the feasibility condition gives 𝑥𝑥_𝐵𝐵+ 𝑥𝑥_𝑆𝑆 = 200. The core doesn’t help us narrow down the set of possible outcomes much at all for this problem. Any division of the 200 surplus with 𝑥𝑥_𝐵𝐵 ≥ 0 and 𝑥𝑥_𝑆𝑆 ≥ 0 is in the core. In other words, the item can trade for any price between 100 and 300.

Now let’s add another seller. The item is worth 300 to the buyer and is worth 100 to each of the two sellers 𝑆𝑆₁ and 𝑆𝑆₂. Only a coalition that includes both the buyer and a seller can achieve the surplus of 200. The following is the coalitional representation.

𝑉𝑉(𝐵𝐵) = 0 𝑉𝑉(𝑆𝑆1) = 0 𝑉𝑉(𝑆𝑆2) = 0 𝑉𝑉(𝐵𝐵𝑆𝑆1) = 200 𝑉𝑉(𝐵𝐵𝑆𝑆2) = 200 𝑉𝑉(𝑆𝑆1𝑆𝑆2) = 0 𝑉𝑉(𝐵𝐵𝑆𝑆1𝑆𝑆2) = 200

The no-blocking inequalities are:

𝑥𝑥𝐵𝐵 ≥ 0 𝑥𝑥𝑆𝑆1 ≥ 0

𝑥𝑥𝑆𝑆2 ≥ 0

𝑥𝑥𝐵𝐵+ 𝑥𝑥𝑆𝑆1 ≥ 200

𝑥𝑥𝐵𝐵+ 𝑥𝑥𝑆𝑆2 ≥ 200

𝑥𝑥𝑆𝑆1 + 𝑥𝑥𝑆𝑆2 ≥ 0

𝑥𝑥𝐵𝐵+ 𝑥𝑥𝑆𝑆1 + 𝑥𝑥𝑆𝑆2 ≥ 200

As usual, combining the last inequality with the feasibility condition 𝑥𝑥_𝐵𝐵+ 𝑥𝑥_𝑆𝑆₁ + 𝑥𝑥_𝑆𝑆₂ ≤ 200 gives

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Any allocation in the core must have 𝑥𝑥_𝑆𝑆₁ = 0. To see why, since 𝑥𝑥_𝐵𝐵+ 𝑥𝑥_𝑆𝑆₂ ≥ 200, then if we had

𝑥𝑥𝑆𝑆1 > 0, we would have to have that 𝑥𝑥𝐵𝐵+ 𝑥𝑥𝑆𝑆1 + 𝑥𝑥𝑆𝑆2 > 200, which would violate feasibility. A similar argument shows that any allocation in the core must have 𝑥𝑥_𝑆𝑆₂ = 0.

Now, since 𝑥𝑥_𝑆𝑆₁ = 0 and 𝑥𝑥_𝑆𝑆₂ = 0, we must have 𝑥𝑥_𝐵𝐵 = 200 since 𝑥𝑥_𝐵𝐵+ 𝑥𝑥_𝑆𝑆₁+ 𝑥𝑥_𝑆𝑆₂ = 200. In words, all the surplus goes to the buyer. The buyer buys the item for 100 from either seller, giving him the full surplus of 200 and leaving the seller with no surplus.

The economics of this example is easy to understand. If there is only one buyer but there are two sellers, then the two sellers will compete with each other and bid the price down until it falls all the way to their reservation values of 100. Economists have understood for a long time that the concentrated side of the market has all the market power against the competitive side of the market.

Example: House Trading Game

There are four people, each with a house. Player 1’s house is ℎ₁. Player 2’s house is ℎ₂. Player 3’s house is ℎ₃ and player 4’s house is ℎ₄. These are the initial allocations. The players have preferences over the various houses as follows.

Player 1: ℎ₃ ≻ ℎ₂ ≻ ℎ₄ ≻ ℎ₁ Player 2: ℎ₄ ≻ ℎ₁ ≻ ℎ₂ ≻ ℎ₃ Player 3: ℎ₁ ≻ ℎ₄ ≻ ℎ₃ ≻ ℎ₂ Player 4: ℎ₃ ≻ ℎ₂ ≻ ℎ₁ ≻ ℎ₄

Which allocations of houses are in the core? Note that players 1 and 3 can form a coalition, swap houses, and each get his preferred house. Thus, any allocation must give player 1 house ℎ₃ and player 3 house ℎ₁.

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Example: Core Non-Existence

While every allocation in the core is efficient, it may be the case that there are no allocations in the core at all.

Here is a simple example. Suppose that three friends have to live in a dormitory. Let’s call them A, B and C. A single room costs $1500 and a double room costs $2000. The friends don’t care whether they live in a double room or a single room; each person’s only goal is to minimize the rent he pays. Interestingly, this simple game has no core allocations.

• It is clearly not in the core for all of them to live in a single room. A coalition of any two roommates can get together in a double room and save money.

• Suppose that two roommates (say A and B) live in a double room and C lives in a single room. At least one of A and B must be paying at least $1000 for the double room rent. Take whichever roommate pays more (or either roommate if they split the rent evenly). Put this roommate in a coalition with C. At the very least, the two of them are together paying $2500 in rent. But they could get together, ignore the third roommate, and reduce their rental costs to $2000.

• Suppose that the roommates live in one double and one single and split the rent jointly between them. At least one subset of two roommates must pay more than $2000. Take these two roommates (if the total rent is split evenly, then take any two roommates). This subset can form a coalition to reduce their rent to $2000 by living together and ignoring the third roommate.

The roommate game has no core allocations. The point is that, although the core has nice efficiency properties, core allocations do not necessarily exist for all games.

Efficiency, Existence and Uniqueness

There are three desirable properties for equilibrium concepts in economics.

• Efficiency – Is the equilibrium outcome in an economic model efficient, in the sense that the total payoff is as high as possible? In other words, is the full surplus realized in any equilibrium outcome?

• Existence – Is there any guarantee that an equilibrium necessarily exists?

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We will consider solution concepts for cooperative games in this context, but it is worth first briefly considering well-known concepts of equilibrium in other economic models to see if they satisfy the desirable properties that we have outlined here.

The table below summarizes results on efficiency, existence and uniqueness for Nash Equilibrium in noncooperative games and, for some perspective, competitive equilibrium in a market economy

Solution Concept Efficiency? Existence? Uniqueness?

Nash Equilbrium in noncooperative games

No – Nash Equilibrium can be inefficient, e.g. prisoners’ dilemma

Yes – Nash (1950), for any finite game (players and strategy set)

No – Many games have multiple equilibria, e.g. coordination game

Competitive Equilibrium in market economies

Yes – First welfare theorem (but possible to have market failure)

Yes – As long as production sets and preferences are convex

No, in general – Yes if all goods in consumption sets are gross substitutes

The potential nonefficiency of equilibrium in noncooperative games, in contrast with the efficiency of competitive equilibrium, has historically been a primary motivation for studying game theory.

Back to cooperative games, what about the core as a solution concept?

• Efficiency? Yes. The no-blocking inequality for the grand coalition ensures that the total surplus available is fully realized in any core allocation.

• Existence? No, in general. The roommate game that we considered above has no core allocations.

• Uniqueness? No, in general. The market example with one buyer and one seller has a whole range of possible core allocations.

So, while the core has nice efficiency properties, there is no guarantee that core allocations actually exist, and when they do exist they are not necessarily unique.

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Shapley Value: Example 1

Consider the following game, expressed in coalitional form.

𝑉𝑉(1) = 0 𝑉𝑉(2) = 0 𝑉𝑉(3) = 0 𝑉𝑉(12) = 2 𝑉𝑉(13) = 1 𝑉𝑉(23) = 3 𝑉𝑉(123) = 5

The no-blocking inequalities that characterize the core, combined with the feasibility condition are:

𝑥𝑥1 ≥ 0 𝑥𝑥2 ≥ 0 𝑥𝑥3 ≥ 0 𝑥𝑥1+ 𝑥𝑥2 ≥ 2 𝑥𝑥1+ 𝑥𝑥3 ≥ 1 𝑥𝑥2+ 𝑥𝑥3 ≥ 3 𝑥𝑥1+ 𝑥𝑥2+ 𝑥𝑥3 = 5

If we reduce the inequalities by substituting from the last equality, we obtain:

𝑥𝑥1+ (5 − 𝑥𝑥1− 𝑥𝑥3) ≥ 2 ⇒ 𝑥𝑥3 ≤ 3 𝑥𝑥1+ (5 − 𝑥𝑥1− 𝑥𝑥2) ≥ 1 ⇒ 𝑥𝑥2 ≤ 4 𝑥𝑥2+ (5 − 𝑥𝑥1− 𝑥𝑥2) ≥ 3 ⇒ 𝑥𝑥1 ≤ 2

Combining with the first three inequalities, the core consists of all allocations that satisfy 𝑥𝑥₁ ∈

[0,3], 𝑥𝑥₂ ∈ [0,4] and 𝑥𝑥₃ ∈ [0,3] and such that 𝑥𝑥₁+ 𝑥𝑥₂+ 𝑥𝑥₃ = 5. This example demonstrates the non-uniqueness issue with the core as a solution concept. There are many allocations in the core.

The idea of the Shapley value is to find the marginal contribution of each party to various coalitions. For example, consider player 1. He could be involved in coalitions of size 1, 2 or 3. Let us find his marginal contribution to each coalition.

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• Size 2: The coalition {12} has value 2, but would have value 0 if player 1 withdrew. Player 1’s marginal contribution is 2. The coalition {13} has value 1, but would have value 0 if player 1 withdrew. Player 1’s marginal contribution is 1.

• Size 3: The coalition {123} has value 5, but would have value 3 if player 1 withdrew and the remaining coalition was {23}. Player 1’s marginal contribution is 2.

The Shapley value is a weighted average of these marginal contributions. The weight for coalitions

of each size are equal. So coalitions of size 1, 2 and 3 each have 1

3 weight. When there are multiple

coalitions for each coalition size, the weight is split evenly among them. For this example,

coalitions of size 2 have 1

3 weight, but there are two of them, so each one has 1

6 weight.

Altogether, the Shapley value for player 1 averages his marginal contributions using these weights.

𝜙𝜙1 =1_{3 ⋅ 0 +}1_{6 ⋅ 2 +}1_{6 ⋅ 1 +}1_{3 ⋅ 2 = 1.17}

Now consider the coalitions in which player 2 can be involved.

• Size 1: The coalition {2} has 0 value, so the marginal contribution of player 2 is 0.

Weighting the size 1 and size 3 coalitions 1

3 and each of the size 2 coalitions 1

6 gives player 2’s

Shapley value:

𝜙𝜙2 =1_{3 ⋅ 0 +}1_{6 ⋅ 2 +}1_{6 ⋅ 3 +}1_{3 ⋅ 4 = 2.17}

Finally, consider all the coalitions in which player 3 can be involved.

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Weighting the size 1 and size 3 coalitions 1

3 and each of the size 2 coalitions 1

6 gives player 3’s

Shapley value:

𝜙𝜙3 =1_{3 ⋅ 0 +}1_{6 ⋅ 1 +}1_{6 ⋅ 3 +}1_{3 ⋅ 3 = 1.67}

The Shapley value allocates the surplus (𝜙𝜙₁, 𝜙𝜙₂, 𝜙𝜙₃) = (1.17,2.17,1.67), which sums to the total available surplus of 5. This is always true. The Shapley value gives a unique rule for allocation of the total surplus available to the grand coalition.

For this example, the Shapley value is in the core, although this need not be the case always.

Shapley Value: Example 2

Consider the following cooperative game, expressed in coalitional form.

𝑉𝑉(1) = 1 𝑉𝑉(2) = 2 𝑉𝑉(3) = 3 𝑉𝑉(12) = 4 𝑉𝑉(13) = 5 𝑉𝑉(23) = 6 𝑉𝑉(123) = 7

The no-blocking inequalities that characterize the core, combined with the feasibility condition, are as follows.

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The core of this game is empty. There are no allocations in the core. To see this, the no-blocking inequality 𝑥𝑥₂ + 𝑥𝑥₃ ≥ 6 together with the feasibility constraint 𝑥𝑥₁+ 𝑥𝑥₂+ 𝑥𝑥₃ = 7 imply that 𝑥𝑥₁ =

1. We can substitute this into the inequalities for the other two-person coalitions:

1 + 𝑥𝑥2 ≥ 4 ⇒ 𝑥𝑥2 ≥ 3 1 + 𝑥𝑥3 ≥ 5 ⇒ 𝑥𝑥3 ≥ 4

But this is impossible. If 𝑥𝑥₁ = 1, we cannot have 𝑥𝑥₂ ≥ 3 and 𝑥𝑥₃ ≥ 4 since the total surplus available is only 7. The core of this game is empty. There are no allocations that satisfy the no-blocking inequalities that characterize the core.

What about the Shapley value? The table below summarizes the marginal contribution of each player to coalitions of various sizes. Again, the idea is to find the marginal addition for a coalition with a player inside the coalition versus a coalition with the player withdrawn.

For example, the coalition {12} has a value of 4, but if player 1 withdrew then what would be left is the coalition {2} which has a value only of 2. Thus, player 1’s marginal contribution is 2.

Player Size 1 coalitions Size 2 coalitions Size 3 coalitions

1 {1}: 1 {12}: 2

{13}: 2 {123}: 1

2 {2}: 2 {12}: 3

{23}: 3 {123}: 2

3 {3}: 3 {13}: 4

{23}: 4 {123}: 3

To calculate the Shapley value, we weight coalitions of each size with 1

3 weight. Because there are

two coalitions of size 2, the weight is split evenly, so each has weight 1

6.

𝜙𝜙1 =1_{3 ⋅ 1 +}1_{6 ⋅ 2 +}1_{6 ⋅ 2 +}1_{3 ⋅ 1 = 1.33}

𝜙𝜙2 =1_{3 ⋅ 2 +}1_{6 ⋅ 3 +}1_{6 ⋅ 3 +}1_{3 ⋅ 2 = 2.33}

𝜙𝜙3 =1_{3 ⋅ 3 +}1_{6 ⋅ 4 +}1_{6 ⋅ 4 +}1_{3 ⋅ 3 = 3.33}

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This example shows that the Shapley value need not be in the core. For example, players 1 and 2 achieve a joint surplus of 3.67 in the Shapley value. But they could break off to form their own coalition and achieve a joint surplus of 4.

Using the Shapley Value to Allocate Costs

A common application of the Shapley value is to allocate costs for a joint project. The idea is the same – to figure out each player’s marginal addition to the cost of various coalitions.

Here is an example. Suppose that four companies want to build an airport. Corporate Jet Company (C) needs a 2000-foot runway that can be built for $2 million. Narrow Jet Company (N) needs a 6000-foot runway that can be built for $6 million (obviously Corporate Jet could use this runway as well). Wide Jet Company (W) needs an 8000-foot runway that would cost $8 million. Finally, Jumbo Jet Company (J) needs a 10,000-foot runway that would cost $10 million, and which in turn could be used by all the other companies.

If the companies agree that it is profitable to build the 10,000-foot runway, how should they divide the $10 million cost among themselves?

The idea is to figure out the addition to the cost by each company for the various coalitions. The simplest case is Corporate Jet Company. He adds $2 million of costs relative to no airport (i.e. the one-person coalition), but he adds nothing in costs for coalitions with any of the other companies.

But consider Jumbo Jet Company. In a coalition with Corporate Jet Company, the airport would cost $10 million for the long runway that Jumbo Jet requires. But Corporate Jet Company alone would require only a $2 million runway. So the marginal addition to cost by Jumbo Jet for this coalition is $8 million.

Company Size 1 coalitions Size 2 coalitions Size 3 coalitions Size 4 coalitions

C {𝐶𝐶}: 2

{𝐶𝐶𝐶𝐶}: 0 {𝐶𝐶𝐶𝐶}: 0 {𝐶𝐶𝐶𝐶}: 0

{𝐶𝐶𝐶𝐶𝐶𝐶}: 0 {𝐶𝐶𝐶𝐶𝐶𝐶}: 0

{𝐶𝐶𝐶𝐶𝐶𝐶}: 0 {𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶}: 0

N {𝐶𝐶}: 6

{𝐶𝐶𝐶𝐶}: 4 {𝐶𝐶𝐶𝐶}: 0

{𝐶𝐶𝐶𝐶}: 0

W {𝐶𝐶}: 8

J {𝐶𝐶}: 10

(14)

The Shapley value places 1

4 weight on coalitions of each size. Since there are three each of size 2

and size 3 coalitions, the 1

4 weight is divided evenly, giving each one a weight of 1 12.

𝜙𝜙𝐶𝐶 = 1_{4 ⋅ 2 +}_{12 ⋅ 0 +}1 _{12 ⋅ 0 +}1 _{12 ⋅ 0 +}1 _{12 ⋅ 0 +}1 _{12 ⋅ 0 +}1 _{12 ⋅ 0 +}1 1_{4 ⋅ 0 = 0.5}

𝜙𝜙𝑁𝑁= 1_{4 ⋅ 6 +}_{12 ⋅ 4 +}1 _{12 ⋅ 0 +}1 _{12 ⋅ 0 +}1 _{12 ⋅ 0 +}1 _{12 ⋅ 0 +}1 _{12 ⋅ 0 +}1 1_{4 ⋅ 0 = 1.83}

𝜙𝜙𝑊𝑊 =1_{4 ⋅ 8 +}_{12 ⋅ 6 +}1 _{12 ⋅ 2 +}1 _{12 ⋅ 0 +}1 _{12 ⋅ 2 +}1 _{12 ⋅ 0 +}1 _{12 ⋅ 0 +}1 1_{4 ⋅ 0 = 2.83}

𝜙𝜙𝐽𝐽 = 1_{4 ⋅ 10 +}_{12 ⋅ 8 +}1 _{12 ⋅ 4 +}1 _{12 ⋅ 2 +}1 _{12 ⋅ 4 +}1 _{12 ⋅ 2 +}1 _{12 ⋅ 2 +}1 1_{4 ⋅ 2 = 4.83}

(15)

Problems

1. Consider the following three-player game represented in coalitional form.

𝑉𝑉(1) = 𝑉𝑉(2) = 𝑉𝑉(3) = 0 𝑉𝑉(12) = 1

𝑉𝑉(13) = 3 𝑉𝑉(23) = 2 𝑉𝑉(123) = 3

a. Write out the inequalities that characterize the core.

b. Reduce the inequalities to find the unique core allocation for this game. c. Find the Shapley value.

2. Sophie (𝑆𝑆) has a choice. Her children are Eva (𝐸𝐸) and Jan (𝐶𝐶). If the three of them live together, then their household surplus is 600. Sophie is depressed living without her children and her payoff alone is −60. The children attain a payoff of 0 living alone, but could run away together and join the circus to attain a joint surplus of 240. Sophie could create a household with just one of her children, but she likes her son better than her daughter. The household surplus in a household with Sophie and Jan is 420, but the household surplus in a household with Sophie and Eva is 300.

a. Represent this game in coalitional form.

b. Describe the allocations �𝑥𝑥_𝑆𝑆, 𝑥𝑥_𝐸𝐸, 𝑥𝑥_𝐽𝐽� in the core.

c. Find an allocation in the core or else show that the core is empty. d. How would the household surplus be divided using the Shapley value?

e. Suppose that Sophie’s depression without her children worsens, so that this payoff falls much lower than −60. How would Sophie’s allocation in (d) change? Explain intuitively without doing any calculations.