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I am not the right person to write about Lloyd Shapley. I think I only saw him once, in the first stony brook conference I attended. He reminded me of Doc Brown from Back to The Future, but I am not really sure why. Here are links to posts in The Economist and NYT following his death.

Shapley got the Nobel in 2012 and according to Robert Aumann deserved to get it right with Nash. Shapley himself however was not completely on board: “I consider myself a mathematician and the award is for economics. I never, never in my life took a course in economics.” If you are wondering what he means by “a mathematician” read the following quote, from the last paragraph of his stable matching paper with David Gale

The argument is carried out not in mathematical symbols but in ordinary English; there are no obscure or technical terms. Knowledge of calculus is not presupposed. In fact, one hardly needs to know how to count. Yet any mathematician will immediately recognize the argument as mathematical…

What, then, to raise the old question once more, is mathematics? The answer, it appears, is that any argument which is carried out with sufficient precision is mathematical

In the paper Gale and Shapley considered a problem of matching (or assignment as they called it) of applicants to colleges, where each applicant has his own preference over colleges and each college has its preference over applicants. Moreover, each college has a quota. Here is the definition of stability, taken from the original paper

Definition: An assignment of applicants to colleges will be called unstable if there are two applicants ${\alpha}$ and ${\beta}$ who are assigned to colleges ${A}$ and ${B}$, respectively, although ${\beta}$ prefers ${A}$ to ${B}$ and ${A}$ prefers ${\beta}$ to ${\alpha}$.
According to the Gale-Shapley algorithm, applicants apply to colleges sequentially following their preferences. A college with quota ${q}$ maintains a waiting list’ of size ${q}$ with the top ${q}$ applicants that has applied to it so far, and rejects all other applicants. When an applicant is rejected from a college he applies to his next favorite college. Gale and Shapley proved that the algorithm terminates with a stable assignment.

One reason that the paper was so successful is that the Gale Shapley method is actually used in practice. (A famous example is the national resident program that assigns budding physicians to hospitals). From theoretical perspective my favorite follow-up  is a paper of Dubins and Freedman “Machiavelli and the Gale-Shapley Algorithm” (1981): Suppose that some applicant, Machiavelli, decides to cheat’ and apply to colleges in different order than his true ranking. Can Machiavelli improves his position in the assignment produced by the algorithm ? Dubins and Freedman prove that the answer to this question is no.

Shapley’s contribution to game theory is too vast to mention in a single post. Since I mainly want to say something about his mathematics let me mention Shapley-Folkman-Starr Lemma, a kind of discrete analogue of Lyapunov’s theorem on the range of non-atomic vector measures, and KKMS Lemma which I still don’t understand its meaning but it has something to do with fixed points and Yaron and I have used it in our paper about rental harmony.

I am going to talk in more details about stochasic games, introduced by Shapley in 1953, since this area has been flourishing recently with some really big developments. A (two-player, zero-sum) stochastic game is given by a finite set ${Z}$ of states, finite set of actions ${A,B}$ for the players, a period payoff function ${r:Z\times A\times B\rightarrow [0,1]}$, a distribution ${q(\cdot|z,a,b)}$ over ${Z}$ for every state ${z}$ and actions ${a,b}$, and a discount factor ${0<\beta<1}$. At every period the system is at some state ${z\in Z}$, players choose  actions ${a,b}$ simultaneously and independently. Then the column player pays ${r(z,a,b)}$ to the row player. The game then moves to a new state in the next period, randomized according to ${q(\cdot|z,a,b)}$. Players evaluate their infinite stream of payoofs via the discount factor ${\beta}$. The model is a generalization of the single player dynamic programming model which was studied by Blackwell and Bellman. Shapley proved that every zero-sum stochastic game admits a value, by imitating the familiar single player argument, which have been the joy and pride of macroeconomists ever since Lucas asset pricing model (think Bellman Equation and the contraction operators). Fink later proved using similar ideas that non-zero sum discounted stochastic games admit perfect markov equilibria.

A major question, following a similar question in the single player setup, is the limit behavior of the value and the optimal strategies when players become more patient (i.e., ${\beta}$ goes to ${1}$). Mertens and Neyman have proved that the limit exists, and moreover that for every ${\epsilon>0}$ there strategies which are ${\epsilon}$-optimal for sufficiently large discount factor. Whether a similar result holds for Nash equilibrium in ${N}$-player stochastic games is probably the most important open question in game theory. Another important question is whether the limit of the value exists for zero-sum games in which the state is not observed by both players. Bruno Zilloto has recently answered this question by providing a counter-example. I should probably warn that you need to know how to count and also some calculus to follow up this literature. Bruno Zilloto will give the Shapley Lecture in Games2016 in Maastricht. Congrats, Bruno ! and thanks to Shapley for leaving us with some much stuff to play with !

This is the first of a series of posts about stability and equilibrium in trading networks. I will review and recall established results from network flows and point out how they immediately yield results about equilibria, stability and the core of matching markets with quasi-linear utility. It presumes familiarity with optimization and the recent spate of papers on matchings with contracts.

The simplest trading network one might imagine would involve buyers (${B}$) and sellers (${S}$) of a homogenous good and a set of edges ${E}$ between them. No edges between sellers and no edges between buyers. The absence of an edge in ${E}$ linking ${i \in B}$ and ${j \in S}$ means that ${i}$ and ${j}$ cannot trade directly. Suppose buyer ${i \in B}$ has a constant marginal value of ${v_i}$ upto some amount ${d_i}$ and zero thereafter. Seller ${j \in S}$ has a constant marginal cost of ${c_j}$ upto some capacity ${s_j}$ and infinity thereafter.

Under the quasi-linear assumption, the problem of finding the efficient set of trades to execute can be formulated as a linear program. Let ${x_{ij}}$ for ${(i,j) \in E}$ denote the amount of the good purchased by buyer ${i \in B}$ from seller ${j \in S}$. Then, the following program identifies the efficient allocation:

$\displaystyle \max \sum_{(i,j) \in E} (v_i - c_j)x_{ij}$

subject to

$\displaystyle \sum_{j \in S: (i,j) \in E}x_{ij} \leq d_i\,\, \forall i \in B$

$\displaystyle \sum_{i \in B:(i,j) \in E}x_{ij} \leq s_j\,\, \forall j \in S$

$\displaystyle x_{ij} \geq 0\,\, (i,j) \in E$

This is, of course, an instance of the (discrete) transportation problem. The general version of the transportation problem can be obtained by replacing each coefficient of the objective function by arbitrary numbers ${w_{ij}}$. This version of the transportation problem is credited to the mathematician F. J. Hitchcock and published in 1941. Hitchcock’s most famous student is Claude Shannon.

The continuous’ version of the transportation problem was formulated by Gaspard Monge and described in his 1781 paper on the subject. His problem was to split two equally large volumes (representing the initial location and the final location of the earth to be shipped) into infinitely many small particles and then match them with each other so that the sum of the products of the lengths of the paths used by the particles and the volume of the particles is minimized. The ${w_{ij}}$‘s in Monge’s problem have a property since called the Monge property, that is the same as submodularity/supermodularity. This paper describes the property and some of its algorithmic implications. Monge’s formulation was subsequently picked up by Kantorovich and the study of it blossomed into the specialty now called optimal transport with applications to PDEs and concentration of measure. That is not the thread I will follow here.

Returning to the Hitchcock, or rather discrete, formulation of the transportation problem let ${p_j}$ be the dual variables associated with the first set of constraints (the supply side) and ${\lambda_i}$ the dual variables associated with the second or demand set of constraints. The dual is

$\displaystyle \min \sum_{j \in S} s_jp_j + \sum_{i \in B}d_i\lambda_i$

subject to

$\displaystyle p_j + \lambda_i \geq [v_i-c_j]\,\, \forall (i,j) \in E$

$\displaystyle p_j, \lambda_i \geq 0\,\, \forall j \in S, i \in B$

We can interpret ${p_j}$ as the unit price of the good sourced from seller ${j}$ and ${\lambda_i}$ as the surplus that buyer ${i}$ will enjoy at prices ${\{p_j\}_{j \in S}}$. Three things are immediate from the duality theorem, complementary slackness and dual feasibility.

1. If ${x^*}$ is a solution to the primal and ${(p^*, \lambda^*)}$ an optimal solution to the dual, then, the pair ${(x^*, p^*)}$ form a Walrasian equilibrium.
2. The set of optimal dual prices, i.e., Walrasian prices live in a lattice.
3. The dual is a (compact) representation of the TU (transferable utility) core of the co-operative game associated with this economy.
4. Suppose the only bilateral contracts we allow between buyer ${i}$ and seller ${j}$ are when ${(i,j) \in E}$. Furthermore, a contract can specify only a quantity to be shipped and price to be paid. Then, we can interpret the set of optimal primal and dual solutions to be the set of contracts that cannot be blocked (suitably defined) by any buyer seller pair ${(i,j) \in E}$.
5. Because the constraint matrix of the transportation problem is totally unimodular, the previous statements hold even if the goods are indivisible.

As these are standard, I will not reprove them here. Note also, that none of these conclusions depend upon the particular form of the coefficients in the objective function of the primal. We could replace ${[v_i - c_j]}$ by ${w_{ij}}$ where we interpret ${w_{ij}}$ to be the joint gain gains from trade (per unit) to be shared by buyer ${i}$ and seller ${j}$.

Now, suppose we replace constant marginal values by increasing concave utility functions, ${\{U_i(\cdot)\}_{i \in B}}$ and constant marginal costs by ${\{C_j (\cdot)\}_{j \in S}}$? The problem of finding the efficient allocation becomes:

$\displaystyle \max \sum_{i \in B}U_i(\sum_{j: (i,j) \in E}x_{ij}) - \sum_{j \in S}C_j(\sum_{i: (i,j) \in E}x_{ij})$

subject to

$\displaystyle \sum_{j \in S: (i,j) \in E}x_{ij} \leq d_i\,\, \forall i \in B$

$\displaystyle \sum_{i \in B:(i,j) \in E}x_{ij} \leq s_j\,\, \forall j \in S$

$\displaystyle x_{ij} \geq 0\,\, (i,j) \in E$

This is an instance of a concave flow problem. The Kuhn-Tucker-Karush conditions yield the following:

1. If ${x^*}$ is a solution to the primal and ${(p^*, \lambda^*)}$ an optimal Lagrangean, then, the pair ${(x^*, p^*)}$ form a Walrasian equilibrium.
2. The set of optimal Lagrange prices, i.e., Walrasian prices live in a lattice.
3. Suppose the only bilateral contracts we allow between buyer ${i}$ and seller ${j}$ are when ${(i,j) \in E}$. Furthermore, a contract can specify only a quantity to be shipped and price to be paid. Then, we can interpret the set of optimal primal and dual solutions to be the set of contracts that cannot be blocked (suitably defined) by any buyer seller pair ${(i,j) \in E}$.

Notice, we lose the extension to indivisibility. As the objective function in the primal is now concave, an optimal solution to the primal may occur in the interior of the feasible region rather than at an extreme point. To recover `integrality’ we need to impose a stronger condition on ${\{U_i\}_{i \in B}}$ and ${\{C_j\}_{j \in S}}$, specifically, they be ${M}$-concave and convex respectively. This is a condition tied closely to the gross substitutes condition. More on this in a subsequent post.