The Commuting Game. July Abstract. We examine commuting in a game-theoretic setting with a continuum

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1 The Commuting Game Alex Anas y and Marcus Berliant z July 2009 Abstract We examine commuting in a game-theoretic setting with a continuum of commuters. Commuters home and work locations can be heterogeneous. The transport network is arbitrary but exogenous. Tra c speed is determined by link capacity and by local congestion at a time and place along a link, where local congestion at a time and place is endogenous. Cars can catch up with others and slow them down. After formulating a static model, where consumers choose only routes to work, and a dynamic model, where they also choose departure times, we describe and examine existence of Nash equilibrium in both models and show that they di er, so prior claims about equivalence of the two models are false. Optima are shown to exist for both the static and dynamic models and are found to be di erent from the Nash equilibria in the respective models. Then it is shown via the folk theorem that for su ciently large discount factors the repeated dynamic model has as equilibrium any strategy that is achievable in the one shot game with choice of departure times, including the e cient ones. A similar result holds for the static model, restricting to strategies in the one shot game that yield individually rational payo s. Applicability of the anti-folk The authors are grateful to David Boyce, whose address at the 2007 Regional Science Association International North American Meetings in Savannah incited his discussants to write this paper. Gilles Duranton, Jan Eeckhout, Lewis Kornhauser and Bill Neilson contributed interesting comments. Mara Campbell and Bill Stone of the Missouri Department of Transportation helped us to obtain access to the important and abundant data on St. Louis tra c, useful for detection of equilibrium strategies in the repeated commuting game, and for that we are especially grateful. y Department of Economics, University of California - Riverside. z Department of Economics, Washington University, Campus Box 1208, 1 Brookings Drive, St. Louis, MO USA. Phone: (314) , Fax: (314) , berliant@artsci.wustl.edu 1

2 theorem is also discussed. In the end, our results pose a challenge to congestion pricing. Finally, we will examine evidence from St. Louis to determine what equilibrium strategies are actually played. JEL number: R41 Keywords: commuting game, folk theorem 1 Introduction 1.1 Motivation Commuting is a ubiquitous feature of the urban economy. Although the classic literature has answered the basic questions in the eld, such as whether equilibrium commuting patterns are e cient, surprisingly some very important questions remain open. If cars can catch up with each other, what can we say about endogenous equilibrium congestion? Do models without an explicit continuous time clock give us an accurate picture of tra c, in the sense that they can approximate behavior in a truly dynamic model? Finally, and most importantly, what happens to commuting when the situation is repeated daily? Does behavior di er dramatically from that observed in the simple context where the commuters know that they only have to commute once? One shot commuting is the exclusive focus of the extant literature. Beckmann et al. (1956) provide a model of rush hour where ows are constant. They analyze optimum and equilibrium in a stylized model with no explicit time in the model, and with a representative commuter. Vickery (1963, 1969) provided the classical analysis of congestion externalities, pricing, and infrastructure investment. Arnott et al. (1993) examine primarily welfare under various pricing schemes when there is only one route or bottleneck, but allowing elastic trip demand and using continuous time. Tra c does not slow down due to congestion, but rather queues at the bottleneck. In their conclusions (p. 177), they note: In the context of rush hour tra c congestion, for example, models should be developed which derive hypercongestion (tra c- jam situations) from driving behavior, solve for equilibrium on a congested network, and account for heterogeneity among users... This is exactly what we attempt. Although the notation used to describe the models formally is burdensome, where possible we will simply give intuition for the results rather than writing down all the technicalities. We formulate both a static model, where time plays no role, and a dynamic model, where it does. The dynamic model features a uniform time by which commuters must arrive at work if they don t want 2

3 to be subjected to a penalty. We assume that commuters have an inelastic demand for one trip per day to work. Our results and the outline of the balance of the paper are as follows. In the next subsection of the introduction, we detail and preview our results with minimal notation by using the simplest example, a network with two nodes and one link where all commuters live at one node and commute to their jobs at the other. In Section 2, we give our notation and specify the general static (timeless) and dynamic models. At this point, we also prove our basic results for each model: that Nash equilibria in pure strategies exist, that Pareto optima exist, and that these two sets generally di er. In Section 3, we examine whether the static model can be viewed as a reduced form of the dynamic model, where time is explicit. The answer is an emphatic NO. In Section 4, we examine Nash equilibria of each of the models when they are repeated daily. By applying the folk theorem, we nd that the set of equilibria is much larger than in the one shot game, be it static or dynamic. It is of the utmost importance that researchers consider this expansion of the equilibrium set when analyzing their models. The repeated game structure yields many more equilibria, even when the folk theorem does not apply, than the one shot game structure studied in the literature. Finally, Section 5 gives our conclusions. 1.2 Example We begin with a simple example to illustrate how the model works and the intuition behind our results. the interval [0; 1] with nodes 1 and 2. Consider commuters uniformly distributed on Each commuter commutes from node 1 to node 2 each day. For simplicity, we only consider the morning rush hour. Denote the capacity of the link by x 2 R +. Suppose that the time it takes to travel the link at the speed limit is t(1; 2) = 1. In the static model, the travel time is given by 1 if the average number of travellers does not exceed capacity of the road, and by 1 otherwise. This means that if road x link capacity is exceeded, then tra c slows down in proportion to the ratio of excess commuters to capacity, max(1; 1 x ). travel time for a commuter on the link is 2. For example, if x = 1=2, then the There really are no choices here for the commuters or a social planner optimizing e ciency, since the route is xed and the model is static; there are no departure times to be chosen. Now consider a dynamic version of the model. Route choice is still xed, but departure (and consequent arrival) times are a choice variable of the com- 3

4 muters. We model departure times in R +, and we call the required arrival time at the destination node 2 (say 9 AM) A 2 R +. There is no penalty for arriving at work early, but the penalty for arriving at work late is 1. This is mainly for illustration. We shall consider nite arrival penalties for both early and late arrival in the remaining sections. They add some complications. Again, in this simple model there is no route choice. But there is a choice of departure time. First, we illustrate how, for any choice of departure times by all commuters, the time to the destination node 2 can be computed. It is assumed that the latter is minimized by each individual commuter at a Nash equilibrium (given the choices of others), and the social planner maximizes a utilitarian welfare function that is minus the sum of commuting times subject to the arrival constraint. The speed of a particular cohort of commuters who depart at the same time is computed as follows. Take the local density of commuters on the road at a particular place on the route and at a particular time. This local density is computed as the limit of neighborhoods on the road of total commuters in the neighborhood divided by the one dimensional size of that neighborhood. The limit is taken as the length of the neighborhood goes to zero. Then, as in the static model, tra c slows down in proportion to the ratio of excess commuters to capacity. An example will help illustrate. Consider the commuters indexed by [0; 1]. Suppose that all the commuters at 0 depart at time 0, all the commuters at 1 depart at time 1, and so forth. Set the arrival time A = 2. We compute tra c speeds (in this case, the arrival time does not bind). With these departure times, when road capacity is high so that x 1, then capacity does not bind. The unit interval of commuters moves from origin to destination at full speed and perfect synchrony, and the local density of tra c is always 1 except for commuters with labels 0 and 1. The density around them is 1 since there 2 is nobody on one side of them (for example the commuters with label 0 have nobody in front of them). But this does not alter their speed, since they are already at the speed limit. In theory, at least, commuters can catch up with those ahead of them (if the ones behind are travelling faster) and slow down the commuters they have caught. What if x < 1? We consider two simple patterns. First, suppose that commuters depart exactly as in the preceding paragraph. Set the arrival time A = Tra c slows down by a factor of 1 relative to the no congestion x x case; thus, tra c speed for the commuters is uniform at x. It takes 1 time to x 4

5 traverse the link, so the last commuters (labelled 1) reach the destination at The local density of commuters is 1 during the commute. Call this x the congested commuting pattern. Now consider the same general departure pattern as in the preceding paragraph, but with commuters labelled 0 beginning travel at time 0, whereas commuters labelled 1 begin their trip at time 1. So the density of commuters x departing at any time is x. Set the arrival time A = Since local density x is the same as capacity, all commuters travel at the speed limit. Thus, travel time for all commuters is 1. Call this the uncongested commuting pattern. These two simple commuting patterns, or strategy pro les, serve to illustrate the computation of local density and speed. Of course, the local density and speed calculations can be much more complicated in, for example, more complicated commuting networks or for more complicated departure patterns. The simple patterns also serve to illustrate the important role played by arrival time. It is rather evident that for the xed arrival time as speci ed at A = 1 + 1, these strategy pro les are Nash equilibria. Notice that all x commuters reach work by the arrival time A for either pattern, but travel time is longer for the congested commuting pattern. Thus, welfare can di er across dynamic commuting patterns even for this simple example. It is evident that the uncongested commuting pattern Pareto dominates the congested commuting pattern. Consider next the comparison of the static with the dynamic model. The rst pattern, the congested commuting pattern, we study for the case x < 1 seems to be the analog of the static case, since tra c speed is constricted. But the second pattern does not seem to have an analog. Thus, the static and dynamic models have di erent Nash equilibrium predictions. Moreover, if the dynamic analog of the static equilibrium is the congested commuting pattern, it is Pareto dominated by another pattern present in the dynamic model but disallowed by the static model. But we can say more. For example, even in the case where the equilibria of the static and dynamic models appear to be the same, if we average congestion for the dynamic model over time and distances on the link, many times and distances have zero commuters and zero congestion. For instance, this happens at distances along the link in our example that the rst commuters have not yet reached. So aggregating the equilibrium of the dynamic model this way will not generate the static model equilibrium, since the ows in the dynamic model will appear diluted. 5

6 An alternative criterion for comparing the static and dynamic model equilibria is to ask that there be at least one time and a distance along each link such that the equilibrium ows of the models are the same. An example relevant to this idea is to use three identical links in series, so that at any given time in equilibrium commuters in the dynamic model are on at most two of the links, unless the links are operating below capacity. Then there is no time at which the ows on all three links are non-zero simultaneously. In section 3 below, we describe how to extend this example so that there is no equilibrium of the dynamic model even remotely resembling the equilibrium of the static model. We return now, and for the remainder of the introduction, to our basic example with 2 nodes and only 1 link. Consider the repeated commuting game for this example. That is, the dynamic commuting game that we have speci ed is played daily. What payo s are attainable? We shall apply a folk theorem below, so the set of payo s attainable as Nash equilibria in the repeated game is related to the payo s attainable in the one shot game. Speci cally, for large enough discount factors in the repeated game, all feasible payo s at least as high as the maximin payo for the one shot game (that are not necessarily Nash equilibria of the one shot game) are attainable as Nash equilibria of the repeated game. In fact, we can show that any payo that is feasible in the one shot game can be attained as a Nash equilibrium of the repeated game. This result is achieved by simply computing the maximin payo of the one shot game. It will be 1. Why? Consider one individual in our simple example. The worst case scenario for that individual in the one shot commuting game is that everyone else who lives at the same node blockades them at time zero. That is, the strategy used by everyone else is to depart at time 0. Then local congestion is in nite, so nobody ever reaches the destination or even moves at all, independent of what the commuter in question does (namely, what departure time strategy they follow). With the model speci ed as we have outlined, a Nash equilibrium in pure strategies or an optimum might not exist. So in what follows, for the dynamic model, we must simplify the problem. This is accomplished by using a xed, nite set of possible departure times that divide equally the time scale in the model. When commuters choose a departure time, they are distributed uniformly over the interval with midpoint their chosen departure time, and length equal to the distance between allowable departure times. With this structure, a Nash equilibrium in pure strategies and an optimum exist. More- 6

7 over, the congested and uncongested commuting patterns we have speci ed are Nash equilibria of the model, and the uncongested commuting pattern is Pareto optimal. What follows below just makes the ideas behind our simple example formal and general. 2 The Commuting Model 2.1 The Static Model: Equilibrium and Optimum We consider a game with an atomless measure space (continuum) of players, a nite set of nodes at which the players live or to which they commute, and a nite set of transport links between the nodes with exogenous capacity. To begin, the measure space of commuters is given by (C; C; ) where C is the set of commuters, C is a -algebra on C, and is a positive, non-atomic measure. 1 We assume that singletons of the form fcg for c 2 C are in C; that for all c 2 C, (fcg) = 0; and 0 < (C) < 1. The origins and destinations in the commuting network are given by a nite set of nodes, denoted by m; n = 1; 2; :::N. Let N = f1; 2; :::; Ng. The commuting network itself is given by a nite set of links between nodes. The capacity of any direct link (with no intermediate nodes) between nodes m and n is given by x mn 2 [0; 1], whereas x nn = 1. m and n does not exist, then x mn = 0. If a direct link between nodes What remains is to specify the strategies and payo s of the commuters. In the static game, there is no choice of time of departure or arrival. There is only route choice. To keep the model simple, we shall examine only the morning, not the evening, commute. We assume that each commuter has a xed origin node, and a xed destination node, with inelastic demand for exactly one trip between the origin and destination. Thus, there is an exogenous, measurable origin map O : C! N and an exogenous, measurable destination map D : C! N. A route for commuter c, denoted r, is a vector of length v no less than 1 but no more than N, with O(c) as its rst coordinate and D(c) as its last. In other words, r is an element of O(c) [f?g [ N [ N 2 [ [ N N 2 ] D(c). Let k be the map that projects a vector onto its coordinate k. A commuting length 1 It is obvious to anyone with half a brain that Skorokhod s theorem implies that we could without loss of generality restrict attention to the unit interval with Lebesgue measure, but this would confuse some of our audience, namely the ones with half a brain or less. 7

8 map is a measurable map l : C! f2; 3; :::; Ng. A commuting route structure is a pair (l; R) where l is a commuting length map and R is a measurable map R : C! N l(c) such that almost surely for c 2 C, 1 (R(c)) = O(c) and l(c) (R(c)) = D(c). Given a commuting route structure (l; R), its ow f 2 R N 2 + is given by f(m; n) = (fc 2 C j 9k 2 f1; 2; :::; l(c) 1g with k (R(c)) = m and k+1 (R(c)) = ng) for m; n = 1; 2; :::N. We assume that if a link is operating below capacity, the time cost of a commuter on that link is constant, call it t(m; n) for m; n = 1; 2; :::N. However, if the link is operating above capacity, then the travel time increases in proportion to the excess of commuters above capacity, f(m;n) t(m; n).2 For example, if the number of commuters is x(m;n) twice the capacity of a link, then the travel time is doubled. We ask that the reader bear this special case in mind, since we use it in all of our examples to give concrete intuition. More generally, we can allow tra c to slow down according to any wellbehaved function of the number of commuters at a distance on a link and link capacity. But for simplicity, we specify the function F : R +! R + where F yields the rate at which tra c slows in response to congestion. We f(m;n) x(m;n) assume that F is strictly increasing and continuous. For our special case, F f(m;n) f(m;n). x(m;n) x(m;n) The time cost of a commuting structure (l; R) for commuter c is (l; R; c) = Thus, l(c) 1 X n=1 max(1; F f(n (R(c)); n+1 (R(c))) )t( n (R(c)); n+1 (R(c))) x( n (R(c)); n+1 (R(c))) is the objective or payo function for each commuter. The utilitarian welfare function for the static model is Z U(l; R) = (l; R; c)d(c) C A Nash equilibrium of the static model is a commuting structure (l; R) such that almost surely for c 2 C, there is no route r of length v for commuter c 2 There is an issue of normalization here, namely whether f is divided by t or not. In essence, it depends on whether a link that is twice as long is half as congested for the same number of commuters on the link. This depends on the interpretation of the static model, whether congestion is viewed as a pulse of commuters or whether they are uniformly spread out over the link. In this paper, we take the view that in the model without time, twice as many commuters on a link results in twice the congestion, no matter the length of the link. However, if one takes the view that length of the link matters, the result is simply division of our f by t, and this makes no essential di erence in the the results we obtain. As we show in section 3, interpretation of the static model is di cult. 8

9 such that Xv 1 (l; R; c) > max(1; F n=1 f(n (r); n+1 (r)) ) t( n (r); n+1 (r)) x( n (r); n+1 (r)) Existence of Nash equilibrium in pure strategies can be proved by applying Schmeidler (1973, Theorems 1 and 2). Rosenthal (1973) proves that a Nash equilibrium in pure strategies exists even when there is a nite number of commuters. Next we prove (informally) that an optimum exists. The problem can easily be reduced to optimization of the utilitarian welfare function over a compact set as follows. Notice rst that there is a nite number of types of commuters, de ned by their origin-destination pairs. Instead of using route choice for each commuter, employ as control variables the measure of each type following each route. Thus, the social planner controls a nite number of variables in a compact set using a continuous objective, so a maximum is attained. We note that due to the congestion externality, the Nash equilibria are unlikely to be Pareto (or utilitarian) optimal. To see this informally, consider an example with 3 nodes. All commuters travel between nodes 1 and 3. There is a direct route, and an alternate route that runs via node 2. The alternative route takes longer than the direct route for each xed number of commuters below capacity because it requires a longer distance of travel. Nash equilibrium of this model has the direct route running above capacity, with the indirect route running below capacity, such that the travel time to work for each commuter is the same. To create a Pareto improvement over the Nash equilibrium, simply move some commuters from the direct to the indirect route. The travel time on the indirect route is the same as at the Nash equilibrium, even for the commuters switched to that route, whereas the travel time for those on the direct route decreases. A 2.2 The Dynamic Model: Equilibrium and Optimum The basics of the dynamic model are the same as those for the static model. To di erentiate the notation, we will add dynamic to the names and add time as an argument of functions. In the dynamic model, each commuter chooses both a departure time (from their origin node) and a route. Routes were discussed in the previous subsection. We allow a commuter to depart at any time d 2 [0; T ]. As we shall see shortly, it is important that this set be 9

10 bounded. From the list of everyone s strategies, each commuter can compute with certainty when they will arrive at their destination node. A dynamic commuting route structure is a triple ( d ; l; R) where d : C! [0; T ] is a measurable function giving departure times for all commuters, l is a commuting length map and R is a measurable map R : C! N l(c) such that almost surely for c 2 C, 1 (R(c)) = O(c) and l(c) (R(c)) = D(c). At this juncture, there is an issue concerning the detail in which we model congestion on each link in the dynamic model. The simplest way to model this is to look only at average congestion on a link. More complicated is to assume that as tra c ebbs and ows, the congestion at the end of the link determines tra c speed on the entire link. The most detailed model, that we use, allows cars to catch up with each other over the course of a link. We use the most detailed model, but assuming link capacity is constant across the link. This is without loss of generality, provided that capacity changes only a nite number of times on a link. In that case, we just add more nodes and links with di erent capacities. We shall de ne commuter progress from origin to destination through a di erential equation in distance. But rst we must de ne progress on each component of a route in a dynamic route structure. First, x a dynamic route structure ( d ; l; R). The basic idea is this. From departure time to the end of the rst link, we follow the di erential equation for congestion for the rst link, and then begin on the second link, and so forth. Notice that the total distance on a link is given by the distance travelled with no congestion in the minimal time: Z t(m;n) 0 1d = t(m; n) For notational simplicity, for i = 1; :::; l(c), de ne i (c) to be the time that node i (R(c)) is reached. Evidently, 1 (c) = d (c). Given a dynamic commuting route structure ( d ; l; R), we shall associate with it a function mn ( m (c); ) that gives as its value the distance travelled on link mn by commuter c at time who begins travel on link mn at time m (c). This function is increasing in its second argument but decreasing in its rst argument. Does such a function exist, and is it unique? Take such a function mn. Compute inductively i+1 (c) = ix j (c)+minf 0 > 0 j i (R(c)) i+1 (R(c))( i (c); 0 ) = t( i (R(c)); i+1 (R(c))g j=1 (1) 10

11 We can then compute its ow at time on link mn at distance, called bf : N 2 R 2 +! R +. It is given by Then compute bf(m; n; ; ) = lim!0 (fc 2 C j mn ( m (c); ) 2 ( ; + )g) 2 (2) _ mn ( m (c); ) = min( F 1 bf(m;n;;mn( m(c);)) x(m;n) ; 1) (3) This describes the progress made by commuters on each link of the entire dynamic commuting route structure for any time. Unfortunately, as mentioned in Section 1.2 above, the system de ning mn, namely (1), (2), and (3), is technically challenging. The reason is that we cannot restrict d, the function de ning the departure strategies of players, beyond assuming that it is a measurable function. Each individual makes a choice, and this is not necessarily coordinated. Discontinuities in departure ows or densities can result in discontinuities in _ mn that rule out our ability to use standard techniques from the theory of ordinary di erential equations as well as the contraction mapping theorem. Moreover, even if we can retrieve a well-de ned mn for each d function, the issue then becomes the fact that there might not exist a Nash equilibrium in pure strategies, since the space of pure strategies is a continuum. Schmeidler (1973) relies heavily on the fact that the number of pure strategies available to players is nite. We solve both of the problems at once by simplifying the dynamic model. Fix small where T= is an even integer, and de ne the departure strategy space to be f; 3; :::; (T= 1)g. This makes the strategy space nite. We assume that all the commuters who choose say will be randomly and uniformly distributed on (0; 2), those who choose the strategy 3 will be randomly and uniformly distributed on (2; 4), and so forth. Theorem 1: The system of equations (1), (2), and (3) with initial condition mn (; ) = 0 for all m; n = 1; :::; N and 2 R + has a unique solution. Proof: We nd the functions mn and i (c) explicitly. Fix a dynamic commuting route structure ( d ; l; R). Let b d (c; 0 ) = d (c) + 0 where 0 is a random variable uniformly distributed on ( ; ) denote the actual departure time of commuter c, that di ers from the chosen departure time d (c) by at most as described just above. To reduce the notational burden, we shall generally suppress the second argument () in any function b. Then b 1 (c) = b d (c). In general, given b i, we will de ne inductively b i+1. Fix any 11

12 origin node m and destination node n 6= m. On each segment mn, de ning mn (c) = fc 0 2 C j d (c 0 ) = d (c); 1 (R(c)) = 1 (R(c 0 )),:::, i 1 (R(c)) = i 1 (R(c 0 )); i (R(c)) = i (R(c 0 )) = m, i+1 (R(c)) = i+1 (R(c 0 )) = ng the default speed for commuter c is given by S mn (c) = min( F 1 (mn(c))=2 x(m;n) ; 1) At the default speed, intervals of commuters never overlap with each other. When this happens, the time on this link is exactly t(m; n)=s mn (c), so b i+1 (c) = b i (c)+t(m; n)=s mn (c) whereas mn (b i (c); ) = S mn (c)[ b i (c)] where i (R(c)) = m. But there are two other possibilities beyond this rst case. The second case is when commuters using di erent routes blend with each other or separate beginning at a node; this is actually a generalization of the concept of default speed. The third case is if a segment of commuters catches up with another along a link. We consider each of these in turn. The second case that is possible in the model is when commuters using di erent routes blend or separate at a node. For the case where they separate, if they are not combined with commuters using other routes, they move at the default speed on the link. But this is just to give intuition. Formally, de ning m mn(c; ) = fc 0 2 C j b 0 j (c 0 ) 2 (b i (c) ; b i (c)); i (R(c)) = m, i+1 (R(c)) = n; j (R(c 0 )) = m, j+1 (R(c 0 )) = n and j 1 (R(c 0 )) = m 0 g m mn(c; ) = fc 0 2 C j 0 m 0 m(b j 1 (c 0 ); b i (c)) 2 (t(m 0 ; m) ; t(m 0 ; m)); i (R(c)) = m, i+1 (R(c)) = n; j (R(c 0 )) = m, j+1 (R(c 0 )) = n and j 1 (R(c 0 )) = m 0 g the speed of commuters is given by S mn(c) = min( F 1 P m 0 6=m lim ( m 0 mn (c;))!0 R m 0 mn (c;) S m 0 m (c0 )d x(m;n)!; 1) = min( F 1 P m 06=m lim!0 ( m 0 mn (c;)) x(m;n) ; 1) Provided that they don t catch up with anyone else, their time on the link is exactly t(m; n)=s mn(c), so b i+1 (c) = b i (c)+t(m; n)=s mn(c) whereas mn (b i (c); ) = 12

13 S mn (c) [ b i (c)] where i (R(c)) = m. This is actually the most general form of the speed and time functions. Notice that since the number of types is nite, the denominators of the right hand side of the last equations actually are almost surely constant for su ciently small. On each segment mn, we say that commuter c catches up with commuter c 0 on link mn if i (R(c)) = j (R(c 0 )) = m, i+1 (R(c)) = j+1 (R(c 0 )) = n b j (c 0 ) < b i (c) t(m; n) S mn(c) S mn(c 0 ) < b i(c) b j (c 0 ) If a set of commuters of positive measure ever catches up with another on a link, an atom accumulates quickly at the upper end of the faster commuter s interval (and at the bottom end of the slower commuter s interval). The slower commuter, who is una ected, continues on at the same speed as before the faster one catches them. The faster commuter literally comes to a stop at that point in the segment, and no commuter who follows on this segment makes any further progress. For any commuter who arrives at this spot afterward on this link, time on this link is in nite. If commuter c catches up with commuter c 0 on link mn, de ne the catch up 3 time, for i (R(c)) = j (R(c 0 )) = m, i+1 (R(c)) = j+1 (R(c 0 )) = n, as = b i (c) + S mn (c0 )[b j (c) b i (c 0 )]. Thus, for all c 00 2 C with Smn (c) S mn (c0 ) k (R(c 00 )) = m, k+1 (R(c 00 )) = n, b k (c 00 ), then b k+1 (c 00 ) = 1 whereas mn (b k (c 00 ); ) = mn (b i (c); ) for all b k (c 00 ) + mn(b i(c); ) is constant for Smn (c00 ) anyone who reaches this spot after the time where the atom forms. The time cost of a dynamic commuting structure ( d ; l; R) for commuter c is R [b l(c)(c; ) ( d 1 (c)+)] d. 2(fc 0 2Cj 1 (R(c))= 1 (R(c 0 )), 2 (R(c))= 2 (R(c 0 )), d (c)= d (c 0 )g) We x an arrival time at A 2 [0; 1]. Next we introduce the arrival penalty function P : R +! R +. To give intuition, think of = l(c) (c). It is given by P () 0 where P ( A ) = 0 For example, in the introduction we required that: Almost surely for c 2 C, b l(c) (c) A Thus, P () = 0 for A whereas P () = 1 for > A. It is actually more common in the literature to use an asymmetric linear penalty function; 3 Also known as catsup. 13

14 see Arnott et al (1993). We can allow further generalization, for example heterogeneous arrival times A, but at the cost of messier notation. We note that in the framework with a nite number of departure times, this is actually the expected penalty for the given choice of strategy, since commuters are randomly assigned over a small interval. With a linear penalty function, this makes no di erence. The individual payo function for the dynamic model is thus: Z [b l(c) (c; 0 ) ( d (c) + 0 ) + P (b l(c) (c; 0 ))] 1 2 (fc 0 2 C j 1 (R(c)) = 1 (R(c 0 )), 2 (R(c)) = 2 (R(c 0 )), d (c) = d (c 0 )g) d 0 The utilitarian welfare function for the dynamic model is Z Z C U(l; R) = [b l(c) (c; 0 ) ( d (c) + 0 ) + P (b l(c) (c; 0 ))] 1 2 (fc 0 2 C j 1 (R(c)) = 1 (R(c 0 )), 2 (R(c)) = 2 (R(c 0 )), d (c) = d (c 0 )g) d 0 d(c) A Nash equilibrium of the dynamic model is a dynamic commuting structure ( d ; l; R) such that almost surely for c 2 C, there is no route r of length v and departure time d0 for commuter c such that, computing arrival times b 0 as in Theorem 1 for the new route and departure time, Z [b l(c) (c; 0 ) ( d (c) + 0 ) + P (b l(c) (c; 0 ))] 1 2 (fc 0 2 C j 1 (R(c)) = 1 (R(c 0 )), 2 (R(c)) = 2 (R(c 0 )), d (c) = d (c 0 )g) d 0 > Z [b 0 v(c; 0 ) ( d0 + 0 ) + P (b 0 v(c; 0 ))] 1 2 (fc 0 2 C j 1 (R(c)) = 1 (R(c 0 )), 2 (R(c)) = 2 (R(c 0 )), d (c) = d (c 0 )g) d 0 We note that due to the congestion externality, the Nash equilibria are unlikely to be Pareto (or utilitarian) optimal. The 3 node example from the end of section 2.1 can be modi ed in an obvious way to account for this as follows. Use a uniform distribution of departures on both the short and long routes, with more people using the shorter route. Simply make the arrival time bind (with in nite penalty) on the shorter, congested route, and few enough commuters on the uncongested, longer route so that the payo s are the same 14

15 on both routes and thus this structure is a Nash equilibrium. The analogous Pareto improvement works, more precisely: move some commuters from the shorter congested route to the longer uncongested route. At this point, there is an important but technical issue that must be addressed. One of the requirements of Schmeidler s results is that utility is continuous (in the weak topology on L 1 ) in the strategy pro le of all commuters. For the model we have stated, this is in fact not true. In all of these models, there is an important discontinuity that must be addressed. A simple version of it can be seen in the example with two nodes and one link. Suppose that a large, slow cohort (exceeding the road capacity) leaves early. Then suppose a small, fast moving cohort leaves soon after, so that the second cohort catches up with the rst along the link. Then when the second cohort catches the rst, it forms an atom at that point along the link and comes to a stop. Suppose that the payo from never arriving is very low. This will hold as long as the second cohort has positive measure. But as we reduce the measure of this second cohort down to zero, it has a very low payo until the limit, where the cohort has zero measure, and catching up with the rst cohort does not result in a penalty. In fact the payo for this zero measure second cohort is larger than for the rst cohort, since the second cohort can leave later but arrive at the same time as the rst cohort. We x this discontinuity in the obvious way as follows. Any strategy pro le that has all strategies used by a positive measure of commuters has its utility for each commuter unchanged. For any strategy pro le that has a strategy or strategies used by only a set of measure zero, for each of these strategies we add a set of measure from outside the model using these strategies (leaving the strategies of the commuters in the model unchanged) and de ne the utility of the limiting strategy to be the limit of the payo s as! 0. With this modi cation, payo functions are continuous. Existence of Nash equilibrium in pure strategies can be proved for the version of the model with discrete and nite departure times by applying Schmeidler (1973, Theorems 1 and 2). For the model with a continuum of departure time strategies, we can only obtain existence of -equilibrium in pure strategies. Similar to the proof for the static model, it is easy to prove that an optimum exists for the discrete departure time model. Instead of looking at a continuum of individual strategies, give the social planner the control variables that are the measure of commuters using each route at each departure time. Given the 15

16 structure in Theorem 1, the utilitarian objective is upper semicontinuous as a function of the measure of commuters using each route and departure time. 4 What does Nash Equilibrium look like in the case of a linear penalty function? This is important for applications, as much of the literature uses such a speci cation. It is actually quite interesting. Suppose that ( ( A ) if A P () = ( A ) if A where, > 0. To x ideas, we consider the example from the introduction, with one link and two nodes, modi ed for this penalty function. At a Nash equilibrium, utility must be equalized across commuters, for otherwise everyone will imitate the happiest ones only. Fortunately for urban economists like us, this is a familiar condition. We claim that Nash equilibrium has the following features. First, nobody arrives late for work. Second, the fastest commuters (smallest cohorts) arrive rst, and su er the greatest early arrival penalty. (For this reason, commuters might not leave at the rst opportunity.) Slower commuters, who depart later with a larger cohort, arrive later, but su er a smaller early arrival penalty. Utility is equalized across commuters. The last cohort arrives at time A. The utility level is determined by the size of the last cohort to depart. Population size, link capacity, link length, and early arrival penalty all contribute to determine the Nash equilibrium of this example. A necessary condition for Nash equilibrium is that it is not possible for a cohort departing very late but travelling at the speed limit to arrive close to A but never catch up with the biggest cohort (that departs last in equilibrium). If smaller cohorts left later than the biggest one, they generally catch up and stop; thus this is not a Nash equilibrium. If larger cohorts left later than the biggest one, they would experience a longer travel time and a late arrival penalty, violating the equal utility constraint; thus this is not a Nash equilibrium. Pareto optimum can feature unequal utility (and thus it is not a Nash equilibrium) where the later departing cohorts have a more even population distribution, and some arrive later than A. 4 In particular, this relies on the strict inequalities in the de nition of catching up in the proof of Theorem 1. 16

17 3 Can the Static Equilibrium be Supported by a Dynamic Equilibrium? Here we ask the following question. Given the identical exogenous data for the static and dynamic commuting games and nding equilibrium, are the ows in the static and dynamic models the same? This is important for addressing the issue of whether the static model makes sense. For if the answer to this question is negative, then there should be no interest in the static model, since its equilibrium behavior is di erent from the analogous dynamic model, and the real world is dynamic. For simplicity, we return now to the examples used in many of the previous sections, namely where there is no penalty for early arrival and an in nite penalty for late arrival. One could imagine that the static model represents some sort of steady state of the dynamic model, where commuters are introduced at constant ows at all the nodes, and the ows in the links are constant over time. But there are two problems with this idea. First, with a xed arrival time (say 9 AM), a steady state does not make sense. The time pro- le of equilibrium departures will generally not be constant over time, since everyone must get to work by the arrival time. Even if arrival time varied, one would not expect to see a steady state necessarily attained. Second, the two alternative concepts for consistency of the two models we introduce next are weaker than asking that a steady state of the dynamic model look like a static equilibrium. In other words, if a steady state of the dynamic model looked like the static model, then the conditions would be satis ed. But they are not. One could ask the question of whether average ows (over time and space or distance on a link) in the dynamic model are equilibrium ows of the static model. Given the identical exogenous data for the static and dynamic games and nding equilibrium, does the following condition on ows hold? 5 f(m; n) = R t(m;n) R A 0 0 bf(m; n; ; )dd t(m; n) A for all m; n = 1; 2; :::; N But this disguises the the following (rather obvious) issue. In the dynamic model, ows could be high for a time and then zero. The average over the link and over time would be in between, but there would be no actual time and 5 In a steady state of the dynamic model, this condition would be satis ed because the ow on each link would be constant, independent of time, and thus be equal to the average ow. 17

18 distance on the link where the average was actually attained. So it is logical to ask whether there is a time, and a distance on every link (m; n), such that the ows from the static model are attained by the dynamic model: 6 f(m; n) = b f(m; n; ; (m; n)) for all m; n = 1; 2; :::; N To answer all of these questions in the negative, one only need go back to the simple example with two nodes and one link given in the introduction. There the uncongested commuting pattern Nash equilibrium is not present for the static model, though the congested one is. But if we want to say something more, for example that there is no equilibrium of the dynamic model that replicates the behavior of the static model, then we must become slightly more sophisticated. As mentioned in the introduction, we set up a network with 3 identical links in series, each one with the structure of the simple example in the introduction (equivalently, one could use 2 nodes and 1 link with the travel time multiplied by 3). Then if we set the arrival time at 1 +3 (where x < 1), the congested commuting pattern violates the arrival time x for the last commuters (they arrive at 3 + 1), and the uncongested commuting x pattern remains as the only equilibrium. It violates all of the conditions above, as there is no uncongested commuting pattern for the static model. In fact, even if we only pay attention to distances on links where there are commuters, their density is x < 1, never to be found in an equilibrium of the static model. 4 The Repeated Commuting Game 4.1 The Commuting Folk Theorem and the Commuting Anti-Folk Theorem It seems obvious that there are few other games better suited to the folk theorem than the (repeated) commuting game, but we have not seen an application. It is not crazy to assume that people play the same game (but not necessarily the same strategy) every day on their commute, modulo random factors such as weather. factors could be added. The folk theorem has proved to be quite robust, so the random In repeated games with a continuum of players, the commuting game is a very nice special case. There are two possible theories of equilibrium behavior, 6 In a steady state of the dynamic model, this condition would be satis ed because the ow on each link would be constant, independent of time, so it would be satis ed for every time. 18

19 both quite famous, namely the Folk Theorem and the Anti-Folk Theorem. The conclusions of the two theorems are in a sense opposites. The rst says that any individually rational, feasible payo s are equilibria of the repeated game. The second says that only Nash equilibria of the one shot game are equilibria of the repeated game. The critical issue in the determination of which one applies is what players observe about other players chosen past strategies in nite repetitions of the game. The formalities can get technical; see Kaneko (1982), Massó and Rosenthal (1989), and Massó (1993). So we describe them in a relatively informal manner. The critical question is this: After nitely many plays of the game, xing one particular individual, can a positive measure of players observe that individual player s behavior? If there is such a set of positive measure for each xed individual, then the folk theorem applies. If no individual s behavior can be detected by a set of players of positive measure, then the anti-folk theorem applies. Note that these two cases are not exhaustive. We shall not take a stand on which theorem applies to our particular case. However, with a nite number of strategies (departure times and routes), it is not far-fetched to think that any particular individual s strategy is observable by those who use the same departure time and route. 7 In the end, which theorem might apply is an empirical matter. There is some evidence that, in other contexts, the folk theorem is relevant; see, for instance, Lee (1999). Let s start by assuming observability and apply the folk theorem. Here we examine two repeated games. The rst has the static model repeated every day, namely a countable in nity of repetitions. The second has the dynamic model repeated every day. The main results, using Kaneko (1982, Propositions 2.1 and 2.1 ), show that if commuters have discount factors suf- ciently close to one, in other words they do not discount the future much, then there is a huge variety of equilibria. The usual folk theorem holds, so any individually rational, feasible strategy (not necessarily a Nash equilibrium in the one shot game) can be obtained as a Nash equilibrium of the repeated game. Kaneko (1982, Proposition 2.1") proves The Perfect Folk Theorem, where we can restrict even to subgame perfect Nash equilibrium and obtain similar results. The equilibrium strategies are supported by various punishment strategies, that apply if the prescribed equilibrium is not followed by a player. Thus, the one day equilibrium is just one of many. Moreover, on the 7 At this point, it is useful to take versions of strategies such that if a set of measure zero plays a particular strategy, then no commuter plays it. 19

20 equilibrium path, one only observes the prescribed equilibrium strategies, not the punishments. Thus, one expects to see the one shot equilibrium played, perhaps, but also (for example) the e cient strategies. For example, in the simple one shot game with two nodes and one link considered in the introduction, if the arrival time is very tight, then only departures at the rst two departure times might satisfy the arrival constraint. If we put few commuters at the rst departure time (exceeding capacity) and the remainder, a larger cohort, at the second departure time (also exceeding capacity), then we will have a Pareto optimum where utility is not the same for all commuters. Thus, it will not be a Nash equilibrium for the one shot game, since the commuters with lower utility will try to imitate those with higher utility. However, it can be supported as a (subgame perfect) Nash equilibrium in the repeated game with discount factor su ciently close to 1. In the static model, the implication is that any feasible route strategy that gives utility at least as high as the maximin payo for the one shot game for each commuter can be achieved as a constant (over time) Nash equilibrium strategy for the in nitely repeated game with no discounting. If we modify this so that the utility of the strategy in the one shot game is at least greater than the maximin utility, then the prescribed strategy can be achieved as a Nash equilibrium strategy in the in nitely repeated game with a discount factor su ciently close to 1. For the repeated dynamic game, assume that lim!1 P () = 1. One strategy producing the maximin payo, equal to 1 for any commuter, is to blockade the commuter at the rst opportunity. If the departure grid is su ciently ne relative to the measure of commuters departing from each origin node, then other commuters can always make any particular commuter arrive as late as desired. Thus, by choosing the grid to be su ciently ne, any payo is above the maximin payo. The upshot is that any feasible departure time and route strategy gives a payo that is at least as high as the maximin payo. So any feasible route and departure time strategy for the one shot game can be supported as a Nash equilibrium of the in nitely repeated game without discounting. If we modify this so that the utility of the prescribed strategy in the one shot game is above 1, then the prescribed strategy can be achieved in the in nitely repeated game with a discount factor su ciently close to 1. Of course, as described above, if no individual s behavior is observable, then for both the static and dynamic models, the only Nash equilibria of the 20

21 repeated game that the Nash equilibria of the one shot game (Kaneko, 1982, Propositions 2.3 and 2.3 ). 4.2 Finite Commuters vs. Continuum of Commuters Here we consider the relevance of models with a continuum of commuters, such as the one we have used. Of course, they are only relevant in the case that they are mathematically convenient approximations to the equilibria of models with a large but nite number of commuters. There will never be data for a world with a continuum of commuters since, in reality, the number of commuters is large but nite. With a nite number of commuters, the anti-folk theorem becomes irrelevant, as the folk theorem applies. Due to this apparent discontinuity in the set of equilibria as the number of commuters tends to in nity, it is imperative to examine the continuity properties of the equilibrium set. Given the discussion of the previous subsection, we consider two cases in the context of the commuting game: when individual strategies are observable and when individual strategies are unobservable. When individual strategies are observable, the commuting folk theorem applies to both the model with a nite number of commuters and a continuum of commuters. Thus, there is no issue of a discontinuity as the number of commuters tends to in nity. When individual strategies are not observable, there is the potential for a discontinuity. The set of equilibria can contract from the set of individually rational, feasible strategies to the set of one shot Nash equilibria. In the model with a continuum of commuters, when an individual commuter changes their strategy, there will be no change in what is observed by other agents, say their commuting time, so there is no basis on which to punish deviators. Thus, the anti-folk theorem applies. But now consider the model with a nite number of commuters. Even if the number of commuters is large, deviations from a prescribed along-the-equilibrium-path-strategy can be detected (for instance by commuters on the same route using the same departure time on the equilibrium path) and therefore can be punished. This explains the contraction of the equilibrium set. However, one can easily argue that as the number of agents gets large, these deviations become undetectable, as their e ects are small and indistinguishable from noise. For example, an analog would be to assume perfect competition in the context of a nite number of agents, where the error from this assumption is small for large economies. 21