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Platooning, driverless cars and ride hailing services have all been suggested as ways to reduce congestion. In this post I want to examine the use of coordination via ride hailing services as a way to reduce congestion. Assume that large numbers of riders decide to rely on ride hailing services. Because the services use Google Maps or Waze for route selection, it would be possible to coordinate their choices to reduce congestion.

To think thorough the implications of this, its useful to revisit an example of Arthur Pigou. There is a measure 1 of travelers all of whom wish to leave the same origin ( ${s}$) for the same destination ( ${t}$). There are two possible paths from ${s}$ to ${t}$. The top’ one has a travel time of 1 unit independent of the measure of travelers who use it. The bottom’ one has a travel time that grows linearly with the measure of travelers who employ it. Thus, if fraction ${x}$ of travelers take the bottom path, each incurs a travel time of ${x}$ units.

A central planner, say, Uber, interested in minimizing total travel time will route half of all travelers through the top and the remainder through the bottom. Total travel time will be ${0.5 \times 1 + 0.5 \times 0.5 = 0.75}$. The only Nash equilibrium of the path selection game is for all travelers to choose the bottom path yielding a total travel time of ${1}$. Thus, if the only choice is to delegate my route selection to Uber or make it myself, there is no equilibrium where all travelers delegate to Uber.

Now suppose, there are two competing ride hailing services. Assume fraction ${\alpha}$ of travelers are signed up with Uber and fraction ${1-\alpha}$ are signed up with Lyft. To avoid annoying corner cases, ${\alpha \in [1/3, 2/3]}$. Each firm routes its users so as to minimize the total travel time that their users incur. Uber will choose fraction ${\lambda_1}$ of its subscribers to use the top path and the remaining fraction will use the bottom path. Lyft will choose a fraction ${\lambda_2}$ of its subscribers to use the top path and the remaining fraction will use the bottom path.

A straight forward calculation reveals that the only Nash equilibrium of the Uber vs. Lyft game is ${\lambda_1 = 1 - \frac{1}{3 \alpha}}$ and ${\lambda_2 = 1 - \frac{1}{3(1-\alpha)}}$. An interesting case is when ${\alpha = 2/3}$, i.e., Uber has a dominant market share. In this case ${\lambda_2 = 0}$, i.e., Lyft sends none of its users through the top path. Uber on the hand will send half its users via the top and the remainder by the bottom path. Assuming Uber randomly assigns its users to top and bottom with equal probability, the average travel time for a Uber user will be $\displaystyle 0.5 \times 1 + 0.5 \times [0.5 \times (2/3) + 1/3] = 5/6.$

The travel time for a Lyft user will be $\displaystyle [0.5 \times (2/3) + 1/3] = 2/3.$

Total travel time will be ${7/9}$, less than in the Nash equilibrium outcome. However, Lyft would offer travelers a lower travel time than Uber. This is because, Uber which has the bulk of travelers, must use the top path to reduce total travel times. If this were the case, travelers would switch from Uber to Lyft. This conclusion ignores prices, which at present are not part of the model.

Suppose we include prices and assume that travelers now evaluate a ride hailing service based on delivered price, that is price plus travel time. Thus, we are assuming that all travelers value time at \$1 a unit of time. The volume of customers served by Uber and Lyft is no longer fixed and they will focus on minimizing average travel time per customer. A plausible guess is that there will be an equal price equilibrium where travelers divide evenly between the two services, i.e., ${\alpha = 0.5}$. Each service will route ${1/3}$ of its customers through the top and the remainder through the bottom. Average travel time per customer will be ${5/3}$. However, total travel time on the bottom will be ${2/3}$, giving every customer an incentive to opt out and drive their own car on the bottom path.

What this simple minded analysis highlights is that the benefits of coordination may be hard to achieve if travelers can opt out and drive themselves. To minimize congestion, the ride hailing services must limit traffic on the bottom path. This is the one that is congestible. However, doing so makes its attractive in terms of travel time encouraging travelers to opt out.

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 !

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