# Connection between Shapley value and Competitive Equlibirium/Payoff

REVISED VERSION. Based on the answer from Holger I. Meinhardt.

I have difficulties understanding the difference between competitive equilibrium of a market game and Shapley values of the coalitional game generated by this market game.

If we don't consider exchange economy but only look at a market game with TU $<{\cal N}, \ell, (w_i), (f_i)>$. We know this market game can generate a coalitional game $<{\cal N},v>$ and its core is not empty. We could investigate this market game from two angles.

(1) Looking at this market game from a decentralized perspective, there exists an equilibrium price $p^*$ and an equilibrium allocation $z^*$. And this pair $(p^*, z^*)$ will generate a payoff profile $\mathbf{x}_{CE} \in R^{|{\cal N}|}$ ($x_{CE}^i= f_i(z^*_i)+p^*(w_i-z^*_i)$. The payoff of agent $i$ is equal to the utility of using $z^*$ goods plus the revenue of selling the rest at a price $p^*$). We know this payoff profile is in the core. We can interpret this as: with all the individual agents try to maximize their own payoffs, collectively they will converge to the core after some time. Then, not only every individual agent has no incentive to leave this equilibrium but also any coalition of agents has no incentive to leave.

(2) Then, let's look at Shapley values associated with the coalitional game $<{\cal N},v>$, which is generated by the market game $<{\cal N}, \ell, (w_i), (f_i)>$. We can calculate Shapley values of this coalitional game $\phi_i(v)$. If we distribute payoffs according to Shapley values, we can guarantee total fairness. These Shapley values will also relate to a certain payoff profile $\mathbf{x}_{SV} \in R^{|{\cal N}|}$ ($x_{SV}^i = \phi_i$).

There is some difference on how to distribute payoffs: in the former case, everyone buys/sells at an equilibrium price, it is a decentralized paradigm. In the latter case, a central authority will allocate the goods to generate maximum payoff and calculate both $\phi_i$ and $f_i$ (the utility of agent $i$ using allocated resource), and eventually remunerate each agent with $\phi_i-f_i$.

My questions are :

• Is the payoff profile $\mathbf{x}_{CE}$ under competitive equilibrium the same as the payoff profile $\mathbf{x}_{SV}$ generated by Shapley values?

• If it is not, can I conclude that $\mathbf{x}_{CE}$ is not a fair payment (because Shapley value is a unique solution)? Even it's not a fair distribution of welfare, everyone will still stick to this equilibrium. Moreover, because CE is Pareto efficient, does it also imply that total fairness cannot co-exist with Pareto efficiency in a general sense?

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Old version(Related to the first answer from Holger I. Meinhardt)

I find difficulties of aligning the following theorems together for the market with transferable utility (TU).

Theorem 1. Competitive equilibrium can generate a competitive payoff profile, which is in the core of the market with TU. [Osborne & Rubinstein Proposition 267.1]

Theorem 2. For a convex game, Shapley value is also in the core. [Osborne & Rubinstein Excercise 295.5]

Theorem 3. There is a unique single-valued solution to TU games satisfying efficiency, symmetry, additivity and dummy. It is what today we call the Shapley value. [Shapley, L. S. (1953). A value for n-person games]

My questions are :

• Can I say the competitive equilibrium (or competitive payoff) of a convex game with TU is equal to Shapley value?

• If this is not the case, does it mean that the competitive equilibrium sometimes is not as good as Shapley value? (In another word, competitive equilibrium cannot guarantee efficiency, symmetry, additivity and dummy, simultaneously)

• Competitive equilibrium is defined in terms of endowments, and endowments are not part of the definition of a TU-game. So talking about " competitive equilibrium (or competitive payoff) of a convex game with TU" makes no sense. – Michael Greinecker Nov 11 '17 at 11:30
• I see. The proposed questions were not well-posed. I have rephrased the questions. Please see the revised version above. – SC.Wang Nov 11 '17 at 15:35

I am not sure if I totally grasp your point. But it seems still to me that you confound both concepts. Of course, one can derive a TU game from an exchange economy, for instance, through

$$v(S) = max \Big[\sum_{i \in S}\, u_{i}(x_{i}) \,\Big\arrowvert\, x = (x_{i})_{i \in S} \in V^{S} \Big] \qquad \forall S \subseteq N.$$

Here one maximizes the sum of utilities of the agents belonging to coalition $S$ under the constraint set $V^{S}$. However, this is only possible if the utilities are of quasi-linear type, otherwise one has an aggregation problem. To overcome such a problem one can rely on a representative selected from the set of agents. This TU game is then a market game derived from an exchange economy, hence a market game is not an exchange economy.

As I mentioned in the first answer, in an exchange economy utilities cannot be transferred. By the preceding maximization, we perceive that utilities are added up, utilities are transferred by the underlying assumption that utilities are quasi-linear. In an exchange economy a Walras equilibrium can be derived under very regular assumptions, in particular, on the utility functions, which need not to be quasi-linear. However, the crucial assumption is that they are not transferable. Hence, a Walras equilibrium or the core (centralized approach) are solution concepts derived within the context of an exchange economy. The Shapley value is a solution of the market game, that is, derived within the context of a transferable utility game. Even though the core exists in the market game, the context is not anymore an exchange economy.

Finally, centralized does not mean that a central planner coordinates the individual decisions by trying to maximize the aggregate utility. It only means that agents can coordinate their actions in order to increase their bargaining power in a stylized bargaining procedure that is hidden behind the characteristic function of the market game. Thus to put it differently, in a market game agents bargain about the proceeds of mutual cooperation, in an economic context normally what the grand coalition (monopoly) can make available. In a market TU game agents can be compensated through transfers to obtain outcomes that could not be achieved without transferable utilities. Thus, the outcome provided by the Shapley value can be interpreted as indicating a bargaining outcome of the market game. Form these considerations it should be clear that the Shapley value is a solution concept for TU games. Of course, there is also an NTU version of the Shapley value, but not anymore under the axiomatization you have listed. By these consideration it should be obvious that the Shapley value of a TU market cannot by compared with the Walras equilibrium of an exchange economy. If so, you have to do it with the NTU Shapley value.

• I am trying digest these remarks part by part. First question, why would generating a coalitional TU game from a market game requires the utility function of each agent to be quasi-linear? I haven't seen this requirement from either textbook [1] or the original paper by Shapley [2]. Could you please point me to some reference? [1] M. Osborne, A. Rubinstein, A Course in Game Theory, p.264. [2] L. Shapley, M. Shubik, On Market Games, 1969, Journal of Economic Theory. – SC.Wang Nov 12 '17 at 15:03
• I suppose that this issue is not broached further is due to the fact that it is considered as common knowledge. Of course, a common knowledge that falls into oblivion. The problem is the aggregation among consumers. See chapter 9.4 in Hal R. Varian, Microeconomic Analyisis (1992). – Holger I. Meinhardt Nov 12 '17 at 15:20
• By not being quasilinear, do you mean there exist some budget constraints on the agent? – SC.Wang Nov 13 '17 at 15:25
• No! For instance, a CES utility function is not quasi-linear. The indirect quasi-linear utility function is of the form $v(\mathbf{p},m)=v(\mathbf{p})+m$. This is a particular type of a Gorman utility function that is given by $v_{i}(\mathbf{p},m)=a_{i}(\mathbf{p})+ b(\mathbf{p})\,m_{i}$. We see that the utility is linear in the income term $m_{i}$, which avoids the mentioned aggregation problem among consumers. – Holger I. Meinhardt Nov 13 '17 at 15:49

You confound the concept of an exchange economy with that of a cooperative game with transferable utilities. The major difference between both is that in an exchange economy utilities are non-transferable and in the latter they are, as the name indicates. In addition, a competitive equilibrium or Walras equilibrium is a decentralized solution concept of general equilibrium theory, whereas the core concept also used therein is a centralized solution concept. In the former case, individual decisions are coordinated by market prices and are therefore decentralized by those. Prices will be influenced by excess demands and not by strategic interaction. In a competitive equilibrium the excess demands are balanced and prices are in equilibrium. The core of an exchange economy, however, is a centralized solution concept whereas individual decisions are coordinated by agents without referring to a price system, only the endowments and preferences matter that lead to excess demands and therefore to a redistribution of the endowments. This can be interpreted as indicating a bargaining process in order to obtain a redistribution of endowments. In the core of an exchange economy no beneficial redistribution of goods among the agents is possible without making at least one agent worse off. Furthermore, in an exchange economy with a countable number of agents, the competitive equilibrium is included in the core. For a continuum of agents both concepts are identical. Indicating that centralized and decentralized decisions lead to same outcome in an environment of perfect information.

Even though the core of a cooperative game (TU or NTU) have similar definitions to those of an exchange economy, they are nevertheless two different mathematical objects and cannot be compared. They are located into different solution spaces. In particular, in a NTU game the endowments are replaced by a characteristic correspondence, i.e., the set of utility allocations that can be achieved by themselves. In contrast, in a TU game the endowments are replaced by a characteristic function instead. The core concept in cooperative game theory with transferable utilities provides -- generally speaking -- a set of outcomes on which no coalition can improve upon while transferring utilities among agents. Such a compensation among agents is not possible in an exchange economy. Finally, it should be obvious, that this is again a centralized approach, since agents can form coalitions to improve their bargaining situation.