میکرو پایه تراکم و قیمت گذاری: چشم انداز تئوری بازی ها

This paper develops congestion theory and congestion pricing theory from its micro-foundations, the interaction of two or more vehicles. Using game theory, with a two-player game it is shown that the emergence of congestion depends on the players’ relative valuations of early arrival, late arrival, and journey delay. Congestion pricing can be used as a cooperation mechanism to minimize total costs (if returned to the players). The analysis is then extended to the case of the three-player game, which illustrates congestion as a negative externality imposed on players who do not themselves contribute to it.

Congestion, one of the most frustrating problems in both freight and passenger transportation, vexes policy-makers, while an understanding of the technical foundations of congestion among policy analysts remains weak. Walters (1961) and Mohring (1970) apply microeconomic theory to congestion, but assume aggregate demand functions and take no account of variation in time or schedule delay (i.e. the journey time is the same for all travelers in the peak). Vickery (1969) and Arnott et al., 1990, Arnott et al., 1993 and Arnott et al., 1998 among others relax that uniform journey time assumption in the bottleneck model, which takes into account time variation as a factor in congestion, and allows travelers to trade-off journey time for schedule delay (see Lindsey and Verhoef (2000) for a summary). These models are extremely useful, but still consider congestion as a product of many travelers. Schelling (1978) argues that macroscopic phenomena should be examined from their micro-foundations, the behaviors of individuals. This paper takes that approach, aiming to build the simplest possible congestion model that reflects real phenomena of schedule delays as well as negative congestion externalities. The paper treats congestion as, at its core, a relatively simple phenomenon with a relatively simple solution. This paper considers congestion as comprising multiple interacting players. The departure time decisions of one commuter affect the journey delay and arrival times experienced by other commuters, leading to interactions and possible gains to all players by cooperating. This paper uses the term journey delay, or travel time in excess of uncongested times, to contrast with schedule delay which refers to the difference in time of departure or arrival compared with preferred conditions. (Other decisions: shifts in mode, route, location, or destination also affect demand of other commuters, but are not included here for clarity of presentation). Game theory, developed by Von Neumann and Morgenstern (1944), presents an analytic approach to explain the choices of multiple actors (agents) in conflict with each other with scope for cooperation, where the payoffs are interdependent (Hargreaves-Heap et al., 1995; Osborne and Rubinstein, 1994; Taylor, 1987). This is distinct from decision theory, where the opponents are states of nature and are passive (Rapoport, 1970). Games are generally classified by the number of players (the games described here are two and three player) and whether the game is zero sum or not. Non-zero sum games engender benefits from cooperation that are absent in zero sum games. Pricing can be a seen as a mechanism to achieve the benefits of cooperation. The application of game theory requires acceptance of certain assumptions about the behavior of actors and their level of knowledge. First, it is assumed that actors are instrumentally rational. Actors who are instrumentally rational express preferences (which are ordered consistently and obey the property of transitivity) and act to best satisfy those preferences. Here it is assumed that travelers minimize total costs (the sum of congestion journey delay penalties, schedule delay penalties [arriving early or late] and prices). Second, it is assumed that there is common knowledge of rationality (CKR). Common knowledge of rationality assumes that each actor is instrumentally rational, and that each actor knows that each other actor is instrumentally rational, and that each actor knows that each other actor knows, and so on. Third, it is assumed that there is a consistent alignment of beliefs (CAB). Each actor, given the same information and circumstances, will make the same decision—no actor should be surprised by what another actor does. Last, it is assumed all players know the rules of the game, including all possible actions and the payoffs of each for every player. This assumption of perfect knowledge, which runs through traditional route choice and congestion pricing models, is strong, and is realistic only in a simple, highly structured game. This assumption is used for expository purposes here, so that the model does not become too complex. Clearly it would be desirable to extend the model to deal with imperfect information, as discussed in the conclusions, though the extent to which that changes the basic results is unknown. In particular, this assumption is more valid for recurring congestion rather than non-recurring congestion (due to incidents, construction, weather, etc.). The congestion game described is a variant of the famous “prisoners’ dilemma” game. A Nash equilibrium is a set of strategies such that no player can improve by changing strategy given that other players keep their strategies fixed. This corresponds very closely with the independently developed Wardrop (1952) User Equilibrium principle used in route choice, and the insight gained from Knight’s (1924) illustration of the equal costs of two used routes. Players’ choose a Nash (user) equilibrium, but if both players were to cooperate, their payoff collectively is higher. A repeated prisoner’s dilemma game may lead to cooperation either implicitly by self-interested players choosing an efficient strategy (e.g., tit-for-tat) or through development of a cooperation-enforcement mechanism. Cooperation in indefinitely repeated games may generate the highest collective payoff (the system optimal solution) (Axelrod, 1984). However because congestion in general is a large multi-driver phenomenon, we expect that cooperation cannot be achieved simply through repetition. A jointly agreed, externally imposed enforcement mechanism would reduce transactions (or bargaining) costs and may achieve a system optimal solution. Game theory has been applied to a number of issues involving transportation such as to help understand airport landing fees (Littlechild and Thompson, 1977), fare evasion and compliance (Jankowski, 1991), truck weight limits (Hildebrand et al., 1990), merging behavior on freeways (Kita et al., 2001), how jurisdictions choose to finance their roads (Levinson, 1999 and Levinson, 2000), aviation (Hansen and Wenbei, 2001), and the political acceptability of road pricing (Marini and Marcucci, 2003). This paper applies game theory to develop the micro-foundations of congestion, and extends the case to congestion pricing. While roadway congestion is normally thought of as a phenomenon involving hundreds or thousands of vehicles, congestion at its most basic, it simply involves two. Examples of significant, time consuming congestion at other transportation facilities that in fact only involve a few (or two) vehicles include ships seeking to use canal locks, airplanes seeking to use airport runways or gates, and trains competing for a single track. Another example is two cars seeking to use a one-car ferryboat to cross a river. In the above situations, those two may want to use a facility that can only accommodate one at any given time, forcing the other to wait. If there were no penalty for arriving early or late, the individuals might coordinate their actions to arrive at different times. However, if there is an advantage to arrive at a particular time (the cost of being early or late exceeds the cost of journey delay), congestion may be a natural consequence. The next section develops a two-player game theoretic congestion model. This is followed by the introduction of pricing to improve the outcome. The subsequent section extends the model to a three-player game. The paper concludes with discussion of applicability of the model and potential extensions.

This paper has developed a new way of viewing congestion and congestion pricing in the context of game theory. Simple interactions among players (vehicles) affect the payoffs for other players in a systematic way. Based on the value of time for various activities (time at origin, at destination, or on-the-road), departure times, and consequently congestion, will vary for the players. Under some range of values congestion will occur between non-cooperative players, even if both would be better off making a different decision. A prisoners’ dilemma is one of many possible outcomes. Congestion is an outcome of individually rational, and sometimes globally rational behavior, under certain preferences. Where there is a difference between the individual and global rational outcomes, congestion pricing can be used, which reduces congestion to an optimal level (but may not eliminate it). The actual early, late, and delay penalties depend on the individual’s value of time, and it is easy to extend the model to handle differentiated values of time. This analysis assumed that players played pure strategies. Extensions to consider mixed strategies (where the play varies from day-to-day with some probability) may yield additional insights. For certain travelers and certain trips, what is most important is not the amount of additional journey time, but instead the certainty associated with arrival time. The importance of this factor depends on the preference function of travelers and the type of trip. In particular, in many cases journey time is affected by the unknown. The randomness in the system and lack of real-time information lead to travel being stochastic. Risk averse travelers may control for this by departing earlier. The relaxation of the assumption of perfect information is an important extension of the model. The use of pricing as an information signal (that few if any actually pay) may be an important result.