William Vickrey is well known for his fundamental contributions to Mechanism Design, including the celebrated second price auction. However, his contributions were not limited to Mechanism Design. In a seminal paper from 1969, Vickrey  identifies bottlenecks as a significant reason for traffic congestion. Bottlenecks are short road segments with a fixed capacity. Once the capacity is reached a queue is formed. Vickrey presents a rush hour model – there are many employees that need to get to work around the same time, all need to cross the same bridge. It is assumed that they have some cost associated with each minute they arrive early to work and a potentially different cost associated with each minute they arrive late to work. Moreover, they have a different (and usually higher cost) for every minute they wait in traffic. Given these costs the employees need to decide when to leave for work in order to minimize their total cost. Similar timing decisions appear in many other situations: e.g., deciding when to go to the doctor or when to enter a traffic intersection.
Vickrey’s paper has inspired a line of work analyzing variants of this model both in economics and transportation theory (for example, [1, 3, 5]). Vickrey, as common in the literature, assumes that the population is continuous. This assumption considerably simplifies the analysis and is motivated by the observation that in large populations the externalities that any single agent is imposing on the rest are negligible. The clear downside of this assumption is that it fails to model scenarios with relatively small population. Moreover, models of discrete and continuous populations can behave differently, as the analysis of the price of anarchy (PoA) in routing games with high-degree polynomials demonstrated. Specifically, while for continuous population the PoA is linear in the degree, it is exponential for discrete population ([29, 28, 4]).
Unlike Vickrey, we study a discrete population model. One of the choices we need to make is whether the agents observe the state of the traffic or not (this makes no difference in Vickrey’s model where the population is modeled as a continuum). The few works that did study discrete variants of Vickrey’s model (for example [23, 26]), all made the assumption that no traffic information is provided (i.e., the queue is unobservable). However, technological advancements such as webcams that are installed over bridges, as well as mobile apps that provide information regarding traffic, call for focusing the analysis around the case that commuters do have some aggregate information on the state of traffic, e.g, they observe the length of the queue. Hence, in our model we assume that the agents do observe the traffic’s state (i.e., observable queue).
Our Model. We study a stylized variant of Vickrey’s model to allow us to focus on two issues that were not explored in the original model: a discrete population and an observable queue.222Our minor simplifications of Vickrey’s model include an assumption that the agents can only join the queue after some starting time.
Formally, we have agents that, starting at time , need to get a service which is offered by a first-come-first-served queue. Time progresses in discrete steps and in each step, each agent needs to decide whether to enter the queue at that time step or stay outside. When multiple agents decide to enter the queue simultaneously, they are ordered by a uniform random permutation. In our model, agents observe everything, and in particular, they observe the length of the queue and the set of agents that are still outside the queue.
For each agent, starting at time , the cost per time step for waiting before joining the queue is normalized to , and the cost per time step for waiting in the queue is (agents dislike waiting in the queue). At each time step a single agent can be processed.
Our model is inspired by traffic related scenarios similar to Vickrey’s rush hour scenario, such as the following one: the first game of the 2018 NBA finals at the Oracle Arena has just ended, and the audience wants to get home to San Francisco. To get home they should all cross the Bay Bridge that has a limited capacity (a bottleneck). While each person wants to get home as soon as possible, he dislikes standing in traffic. So, he should strategically decide when to leave the stadium and try to cross the bridge, aiming to minimize his discomfort. Hanging out around the stadium is an option that is costly, but not as much as standing in traffic. Fortunately for the audience, there are cameras installed over the bridge and apps that constantly broadcast the traffic state, and they can observe it and make their decision accordingly.
Solution Concept. As our game is symmetric and all agents are ex-ante the same, we focus on equilibria in symmetric randomized strategies that are anonymous. In such profiles all agents play the same randomized strategy that does not depend on the identities of the other agents. Observe that as agents are symmetric, equilibrium in asymmetric strategies requires different agents to play differently although they are ex-ante symmetric. This requires the agents to coordinate on which strategy each of them will play. Thus, symmetric strategies are arguably more natural than other, more general, strategies333In , Otsubo and Rapoport are making a similar argument in favor of symmetric strategies..
Moreover, our focus will be on stationary strategies that do not depend on the time step, but rather only on the state – the number of agents that are outside the queue as well as the number of agents that are in the queue. More formally, we consider symmetric Nash equilibria in anonymous stationary strategies, or symmetric strategies
for short. Under such strategies, for any state there is some defined probability of entering the queue, and that probability is used by all agents. In AppendixB we prove that such equilibria always exist.
The discrete population and discrete time assumptions make the analysis of our model quite challenging. In particular computing equilibrium strategies for a game with agents requires solving polynomial equations of degrees increasing from to . Otsubo and Rapoport  are among the few that studied discrete population variants of Vickrey’s model. They have only provided a complicated algorithm to numerically compute symmetric mixed Nash equilibrium rather than obtaining closed-form expressions.
Other approaches that were taken in similar models include analyzing the fluid limit of the system , and computing an equilibrium for the continuous time model by solving differential equations . We take a different approach and instead of explicitly computing a symmetric equilibrium, present asymptotically tight upper and lower bounds on the social cost of any symmetric equilibrium.444Doing so alleviates the need to precisely compute symmetric equilibria and the need to determine if the game has a unique symmetric equilibrium or not.
Our Results. We study the efficiency of symmetric equilibria in terms of the social cost, which is simply the sum of the players’ costs. We present bounds that hold for any symmetric equilibria, any and any .555While we prove bounds that hold for any and any (see Theorems 4.1 and 4.3) our focus in the presentation is on the asymptotic social cost of symmetric equilibria, when either or gets large. Thus, we establish tight asymptotic price of stability and price of anarchy results for symmetric equilibria. Moreover, our lower bounds imply price of anarchy lower bounds for general Nash equilibria. 666Similarly to the “Fully Mixed Nash Equilibrium Conjecture”  we suspect that in our game symmetric equilibria are in fact the worst equilibria.
We first analyze the ratio between the social cost of any symmetric equilibrium and the social cost of the optimal solution. We observe that whenever (the cost of waiting one time step in line is at most twice the cost of waiting outside), the unique symmetric equilibrium is for all agents to enter the queue immediately, and the total social cost is . On the other hand, when agents enter sequentially, the social cost is only (this is also the minimal social cost when we do not impose any restrictions on the strategies, in particular the strategies can be non-anonymous and non-stationary.)
Thus, in this case the ratio between the social costs of the unique symmetric equilibrium and the optimal solution is . Loosely speaking a similar bound of also holds when we take a fixed which is considerably smaller than , but the proof is much more involved. A bit more formally, we prove the following result:
Fix . As approaches infinity the ratio between the social costs of any symmetric equilibrium and the optimal solution is approaching .
Usually in scenarios such as commuting to work or deciding when to head to the bridge after a game, the number of agents is relatively large while the normalized cost of waiting a unit of time in traffic, , is relatively small. Theorem 1.1 tells us that in such cases the loss of efficiency in a symmetric Nash equilibrium is constant and relatively low, only 2.
We next consider the other extreme parameter regime, where the cost is large relative to . Such scenarios might arise in cases where either people do not care so much about getting the service early (for example, taking a routine medical check-up or running some non-urgent bureaucratic errand) or they have an arbitrary high cost for arriving simultaneously with others to receive the service. For example, one can think of a traffic intersection as providing a service for which simultaneous entry might cause an accident and has a very high cost. For the regime that is small relative to we obtain qualitatively different results than those for the regime that is large relative to . This provides an additional confirmation for the value of studying models of discrete population.
Fix . As approaches infinity the ratio between the social costs of any symmetric equilibrium and the (unrestricted) optimal solution is approaching , up to an additive term of .
The theorem shows that when is significantly greater than , the multiplicative efficiency loss is very high (grows asymptotically as ). Essentially, the issue with symmetric equilibria is that when the cost of standing in line is very high, the players are so horrified at the prospect of waiting in the queue that they enter the queue at a very low probability. Thus, the service is actually idle most of the time. To reduce this kind of inefficiency, society came up with symmetry breaking mechanisms such as traffic lights or doctors appointments (see ), which provide a much needed coordination. Such a coordination mechanism induces a sequential order of entry, no player ever waits in the queue and the social cost is minimal. Moreover, for large (), this is actually an (asymmetric) equilibrium.
Theorem 1.2 shows that symmetric equilibria are highly inefficient, but what is the main source of the inefficiency of symmetric equilibria: is it the players’ strategic behavior or is it the lack of coordination imposed by symmetric strategies? As far as we know, no prior work has tried to separate between the loss of efficiency of symmetric equilibria due to strategic behavior and due to the symmetry requirement. To answer this question we bound the cost of the symmetric optimal solution and provide bounds on the ratio between the cost of any symmetric equilibrium and the symmetric optimal solution. We derive two different asymptotic bounds, depending on whether or are fixed.
Fix . As approaches infinity the ratio between the social costs of any symmetric equilibrium and the symmetric optimal solution is approaching .
In fact, as the ratio between the social costs of any symmetric equilibrium and the optimal solution is also approaching , this implies that as grows large the social cost of the symmetric optimal solution is approaching the social cost of the (unrestricted) optimal solution, and both have essentially the same gap from any symmetric equilibrium. Thus, for this case, we conclude that the main source of inefficiency of symmetric Nash equilibria is the strategic selfish behavior of the players.
When considering the regime in which is fixed but large, while grows to infinity, a different picture emerges. Intuitively, since the cost of having two or more agents entering the queue simultaneously is so large, to avoid this cost, any profile of symmetric strategies must use entry probabilities that are low enough to ensure that the expected number of agents that join the queue at each step is much lower than . As a result, the agents will wait for a long time before anyone enters the queue, which implies a high social cost.
This creates a large gap between the social costs of the symmetric optimal solution and the (unrestricted) optimal solution. Interestingly, the gap is of the same magnitude as the gap between the worst symmetric equilibrium and the optimal solution. We show:
Fix . As approaches infinity the ratio between the social costs of any symmetric equilibrium and the symmetric optimal solution is approaching , up to an additive term of .
Recall that Theorem 1.2 is showing that as approaches infinity the ratio between the social costs of any symmetric equilibrium and the optimal solution is approaching . When we combine the two theorems we conclude that in the case that goes to infinity, the main source of inefficiency of the symmetric Nash equilibrium is the lack of coordination in symmetric randomized strategies. Nevertheless, there is still some small constant loss that does not vanish and is due to incentives, yet it dwarfs compared to the loss of (which tends to infinity as grows to infinity) that is due to lack of coordination.
Related Work. Bottleneck models were studied in both the traffic science literature and the economics literature. Arnott et al.  provide economic analysis of Vickrey’s bottleneck model and also consider how tolls can reduce the cost associated with strategic behavior in such models. Later papers extended the model to handle more general pricing schemes (for example [3, 7]). In  Arnott et al. consider giving traffic information to commuters in a continuous population model in which commuters need to choose when to leave and which route to take. The new twist is that travel time is affected by unexpected events such as accidents or bad weather that the commuters can get information about. Arnott et al. reach the conclusion that providing the commuters perfect information about these unpredictable events can eliminate the inefficiency resulting from them. Other variants of the model that were studied more recently include: heterogeneous commuters  and the effects of congested bottlenecks on the roads leading to them .
The literature on strategic queuing is also related to the current paper. Similar to Vickrey’s model, this line of work has also originated from a paper first published in 1969. In his seminal paper, Naor was the first to introduce both economic and strategic considerations into the queuing literature 
. Up till then queuing theory mainly focused on the efficiency of queues. The most well known model in classic queuing theory is the M/M/1 queue model, where there is one server and the jobs arrive according to a Poisson distribution and have an exponentially distributed service time. According to Naor’s model, the service has a price and the “jobs” need to decide whether to join the queue or not. This gave rise to a new area called strategic queuing which studies the users’ behavior in different queuing systems under various assumptions (see[13, 12] for extensive surveys).
In general, the literature on strategic queuing has traditionally focused on models of unobservable queues as these are easier to analyze (see chapter 2 of  for a survey of recent works on observable queues). Hassin and Roet-Green  bridge the gap between observable queues and non-observable queues by presenting and analyzing a natural model in which the agents have the option to pay to see the length of the queue. Most of the works in strategic queuing (both on observable and unobservable queues) consider games in which each player arrives at some time and needs to immediately decide whether to join the queue or not. In this setting, the paper of Kerner  applies a solution concept similar to ours in studying symmetric equilibrium joining probabilities for an M/G/1 observable queue.
The more elaborate model in which the player’s strategy is to choose an arrival time with the goal of minimizing his waiting time was first suggested in  for unobservable queues.  studied a model in which agents can choose to join the queue before its opening time (early arrivals) while later  showed that the efficiency of the equilibrium can be sometimes increased by disallowing early arrivals. Discrete time and discrete population versions of this model were later studied in [22, 27] that concentrated on symmetric mixed Nash equilibria for this unobservable queue model. A recent work  studies a setting in which while the queue is unobservable the service provider can observe the queue and give the agents some information regarding the queue.
A specific line of papers in strategic queuing which is similar both in intuition and in formalism to our model is on the so called “concert queuing game”. This game was first defined in  and was later studied in follow up papers such as [19, 17]. In this game concert attendants wish to get home as soon as possible once the concert ends but they dislike standing in traffic. The main distinctions between the concert queuing game and our model are in the assumptions on the observability of the queue and its processing time (in our model, processing time is fixed, while in the other model it is distributed according to some distribution). In each of  and  the authors use different analysis techniques to establish a bound of on the price of anarchy for their model.
The strategic queuing literature includes a few papers dealing with price of anarchy. The first one was  which studied the price of anarchy of a multi server system where the players strategically choose which queue to join (without observing the queues’ states first). In  Gilboa-Freedman et al. provide a price of anarchy bound for Naor’s model  where the queue is observable but the strategy of each player is limited to the one-time decision whether to join the queue or not.
The paper of Fiat et al.  also considers strategic entry by selfish players – players that need to broadcast on a joint channel. The model in that paper is fundamentally different than ours, as simultaneous entry results in all players failing to enter the channel, rather than a formation of a queue as in our model.
Paper Outline. We start by formally presenting our model and some useful observations. Next, in Section 3 we study the two player case as a warm-up. We then present our upper and lower bounds on the cost of any symmetric equilibria in Section 4, Theorems 1.1 and 1.2 follow from these results. Finally, we present bounds for the social cost of symmetric optimal strategies in Section 5, Theorems 1.3 and 1.4 follow from these bounds.
2 Model and Preliminaries
There is a set of identical agents and time is discrete. We use to denote a time step.
At each time step an agent has two possible actions; enter the queue (1) or wait (0). Agents that did not enter yet are said to be outside the queue, and agents that entered but are still in the queue are in line. An agent that enters at time will be processed after every agent that entered at time . If multiple agents decide to enter at time , they will enter the queue in a uniform random order. More formally, at every time step the following sequence takes place:
Each agent decides whether or not to enter the queue (possibly using randomization).
After the agents make their decisions, all agents that have decided to enter the queue are added to the end of the queue in a random order.
If the queue is not empty then the first agent in the queue is processed. The rest of the agents in the queue incur a cost of .
Every agent outside the queue incurs a cost of .
The goal of each agent is to minimize his expected cost. We use to denote a game with agents and waiting cost per unit of .
In general, a strategy of an agent needs to specify the probability of entry at each history, such a history specifies the time, the realized action of each agent at every prior time, and the randomization results whenever multiple agents that enter at the same time are ordered at the end of the queue. Our focus in this paper is on anonymous stationary strategies – strategies that do not depend on the time step or the identity of agents, such strategies will only depend on a summary statistics specified by the (anonymized) state of our game. A state of our game is defined as a pair , where there are agents that are still outside the queue and agents in the queue, and . We formally define anonymous stationary strategies as follows:
Definition 2.1 (anonymous stationary strategies in )
A strategy of an agent is an anonymous stationary strategy if for any history it specifies a probability that an agent enters the queue that is only a function of the state . We denote that probability of entry at a state by and the probability at state by .
We use to denote a profile of strategies. A profile of anonymous stationary strategies is symmetric if all players use the same strategy ( for every ). We are interested in Nash equilibria of the game, in such equilibria each player minimizes his cost, given the strategies of the others. To be brief we sometimes refer to a “symmetric equilibrium in anonymous stationary strategies” simply as symmetric equilibrium. In Appendix A we discuss general strategies and show that if a profile is an equilibrium with respect to anonymous stationary strategies, it is also an equilibrium with respect to any arbitrary strategies that might depend on the time or on the identities of the agents. For this reason, for the rest of the paper we will only consider deviations to anonymous stationary strategies.
We denote the expected cost of every player in a symmetric equilibrium in the game by . When and are clear from the context we simplify the notation to . We denote the social cost of strategy profile by . As we will see, it is useful to extend this notation to sub-games as well. We denote the sub-game that starts with a state by . For a symmetric equilibrium we denote the cost of each of the players that are outside the queue by . When and are clear from the context we will use the shorter notation . With this notation we have that .
2.1 Basic observations
If at state a player enters with a non-trivial probability then he must be indifferent between entering and waiting at that step. When not indifferent, the player will choose to either enter the queue or wait with probability one. Our analysis of the social cost of symmetric Nash Equilibria heavily depends on this observation. Thus, it is useful to first work out the expressions for a cost of a player that joins the queue with probability at state and the cost of player that joins the queue with probability at state . We define the two costs as and respectively, where we assume that in state all the players but enter with probability and at any other state all the players play according to some symmetric strategy profile . We now give expressions for the two costs for every symmetric profile :
For every , , and : .
Proof: The cost has two parts: a cost associated with players that joined the queue together with player but were randomly assigned to be before him in line, and a cost associated with the players already standing in line. The second cost is simply . The first part of the cost can be computed as follows: for each player
we can define a random binary variablewhich equals if enters the queue before and otherwise. We observe that , since with probability it will enter the line and since the player in the queue are order randomly with probability it will be ordered before player . Now, by linearity of expectation we get that the expected number of players that will be before player in line is .
For every , and :
Proof: It holds that
This is so as the agent pays for waiting, then with probability no other agent enters, and we are back in the same situation, and if exactly other agents enter, an event that happens with probability , the agent will pay since one of the agents that entered will be processed.
Similar computation holds for , with the exception that if no player enters, one of the agents in the non-empty queue will be processed, hence we get the second equation.
Next, we claim that a symmetric equilibrium always exists. This is not a priori clear as our strategies cannot be easily defined as a mixed of pure anonymous stationary strategies.
Fix any and . There exists a symmetric equilibrium in anonymous stationary strategies in the game .
The proof idea is rather simple, we show that a symmetric equilibrium exists in every sub-game , by induction on . Essentially, in every such sub-game we show that it is either the case that there exists such that and hence there exists a random symmetric equilibrium in which agents enter with probability at this state, or for every value of , or in which case there is symmetric equilibrium in which all players either do not enter the queue or all players do enter the queue, at this state. The formal proof is slightly more complicated since the function is not continuous at . The complete proof can be found in Appendix B.
We observe that for small values of () the unique symmetric equilibrium outcome is for all players to enter immediately.777Note that the uniqueness is for every sub-game that is actually played - as all players enter immediately, states are never reached and the play of the game as well as the payoffs are independent of how players intend to play in such sub-games .
For any . If then in the game the unique symmetric equilibrium with anonymous stationary strategies is for all agents to enter with probability and the social cost is .
Proof: We will show a stronger claim: in the game for a player’s dominant strategy is . To this end, it is enough to show that for any number of players that joined the queue in the first step (after the randomness was realized) the player prefers to join the queue with probability in the first step. To see why this is the case, note that if players joined then the player’s cost for joining the queue is . On the other hand, if he chooses not to join the queue his cost will be at least as he will need to wait for at least time steps before leaving the queue.
We focus our equilibrium analysis on the case that . We now show that when it is no longer the case that a player in a symmetric equilibrium prefers to join the queue with probability . Furthermore, in case the queue is empty, deterministically not entering the queue is also not a best response to the strategies of others in a symmetric equilibrium.
For any , and for any such that , and , in any symmetric equilibrium and .
Proof: We show that for such that and , in no symmetric equilibrium it holds that . Indeed, the cost of an agent by entering when all other agents enter with probability is , while if the agent waits till all others are processed and then enter, his cost is only . Thus, in any symmetric equilibrium . The claim that is immediate since if all other players do not enter, a player wants to enter immediately.
The following observation characterizes the cost of symmetric Nash equilibria based on the observations above:
Fix and . Given a symmetric equilibrium in the game , the cost of some player which is outside the queue in the state satisfies the following:
For it holds that .
For , if then . Otherwise, and .
3 Warm-up - The 2 Players Case
To get some intuition, before turning into the general case with many agents it is instructive to consider first the simple case of only players. For this case we obtain an exact expression for the players’ costs in the unique symmetric equilibrium.
Note that a single player will always join the queue () and for this reason . Now, to compute a symmetric equilibrium we only need to compute the probability that the agents enter when both are still outside ().
For players, if , there exists a unique symmetric equilibrium in anonymous stationary strategies in which each player plays the strategy: , . The social cost for both players in this equilibrium is .
Proof: To compute we use Observation 2.7 implying that a player in the game is indifferent between joining the queue with probability and staying outside. By Observation 2.2 the cost of the first option is and by Observation 2.3 the latter option has a cost of , note that this is the expected number of steps till the other player enters, when this player waits outside. Putting this together we get that , and thus . The social cost to both players is .
We compare the cost at Nash equilibrium against the cost of the optimal solution. For two players, the cost of the optimal solution is simply as one of the players will enter first and pay a cost of and the other will enter second and pay a cost of . This implies the following corollary:
In the game the ratio between the social costs of the unique888Note that when the two players game admits exactly three equilibria: the two optimal equilibria in which one player enters after the other, and the symmetric random equilibrium we discussed. Thus, our result is both a price of anarchy result for unrestricted equilibria and a price of stability result for symmetric equilibria. symmetric Nash equilibrium and optimal solution is .
This relatively large gap that grows with leads us to ask what is the source of this gap – is it due to strategic behavior, or to the lack of coordination imposed by symmetric strategies? To answer this question we compute the minimal cost when all agents are required to use the same strategy and use anonymous stationary strategies, which are not necessarily an equilibrium. We note that even just for players computing the optimal symmetric solution is simple yet not completely trivial, as it requires computing the minimum of a function which is the ratio of two polynomials. As the number of players increases this becomes more complicated and hence instead of directly computing the optimal symmetric strategy we will compute bounds on its cost.
As in the Nash equilibrium, once an agent is the only one outside, he clearly enters immediately. Thus, we only need to compute the probability of each agent entering, assuming both agents are outside (denoted ).
For players, if , the symmetric anonymous stationary strategy that minimizes the social cost is: and . The social cost for this profile is .
Proof: Let . It is easy to see that:
Clearly, is not optimal. To find that minimizes this we take the derivative by and check when it equals zero:
The unique solution for this equation in is and it is easy to verify that this is indeed the minimum in . Moreover, the left side derivative at is positive, so is not a minimizer. We conclude that is the unique minimizer of this function in , for any . For this the value of is
After simplifying this expression a bit we get that:
The following corollary is easily derived from the above claim:
In the game the ratio between the social costs of the unique symmetric Nash equilibrium and optimal solution in symmetric strategies is approaching as approaches infinity.
We conclude that for large values of , there is a huge loss for insisting on symmetric profiles: the optimal cost grows from in an asymmetric optimum, to about in the symmetric optimum. An additional, much smaller, loss of factor comes from further requiring the symmetric profile to be an equilibrium. Our goal in this paper is to understand the source of inefficiency of symmetric Nash equilibria for any . We present a separation between the cost ratio of symmetric Nash equilibria and the symmetric optimal solution when is fixed and is large and the case that is fixed but is large.
4 Bounds on the Cost of Symmetric Nash Equilibria
In this section we provide bounds on the cost of symmetric Nash equilibria in any profile of anonymous stationary strategies, these bounds hold for any and any . We present two types of bounds, each will be tight for a different regime of the parameters and , and we use these bounds to prove Theorem 1.1 and Theorem 1.2. We first present a bound that is useful when is relatively small compared to .
For every , and symmetric equilibrium :
Clearly the above bound is asymptotically tight whenever . This implies that when , for a sufficiently large value of , the social cost of any symmetric equilibrium is about . Denote by the social cost of the optimal solution. Recall that in the optimal solution the players enter sequentially and hence the cost of the optimal solution is . The ratio between the cost of any Nash equilibrium and the cost of an optimal solution is essentially , proving Theorem 1.1. Formally:
For every fixed and every there exists such that for any for every symmetric equilibrium it holds that
Next, we give a different bound which will be tight for the case of large enough , and that goes to infinity. We show that the social cost of any symmetric equilibrium is essentially approaching . Formally:
For every , and any symmetric equilibrium :
for some decreasing function that for any fixed , converges to as grows to infinity.
In this case if we get that the social cost of any symmetric equilibrium is about and hence the ratio between the costs of any symmetric equilibrium and the optimal solution (which has cost of ) is approaching , proving Theorem 1.2:
Fix any . There exists such that for any there exist such that for any it holds that for any symmetric equilibrium :
To prove Theorems 4.1 and 4.3 we separately prove an upper bound and a lower bound on the cost of any symmetric equilibrium. We derive the lower bound for Theorem 4.1 by basic observations on our game. Our proof for the lower bound for Theorem 4.3 is much more involved, and it utilizes the upper bound of Theorem 4.3. In the next section we present a general recipe for proving both of our upper bounds. Then, we provide in Section 4.2 a complete proof of Theorem 4.1. The proof of Theorem 4.3 can be found in Section 4.3.
4.1 A General Recipe for Proving Upper Bounds
The proofs of the two upper bounds of Theorem 4.1 and Theorem 4.3 follow a similar structure based on an induction. We define a general notion of a ”nice upper bound function” and show that this notion is useful in bounding the social cost in any symmetric equilibrium without explicitly computing it. Each of our two upper bounds is proven by a using different upper bound function , each exhibits some nice properties captured by the following definition.
A candidate upper bound function is nice if:
For every and : .
is monotone in the following sense: for every and , that is, moving agents from outside the queue to the queue decreases the upper bound on the cost.
Next, we show how the definition of nice candidate upper bound function can assist us in proving upper bounds for our game.
Consider a candidate upper bound function that is nice, and a game such that . Assume that for every symmetric equilibrium it holds that for every , such that , we have that . Then:
For every such that and , it holds that .
Proof: For the first statement, when such that , we know that an agent can always decide to wait for steps till the agents in the queue are being served. Thus, we have that
where the last transition is by using the proposition assumptions. Now, the monotonicity condition on (the second condition of Definition 4.5) comes in handy as it tells us that . The proof is completed by using the first requirement of a nice candidate upper bound function which tells us that .
The proposition now follows from using the fact that for every and such that and the monotonicity of to observe that .
Using Proposition 4.6 as part of an induction on the state space we derive the two upper bounds that hold for any and :
4.2 Bounds for Small (for Theorem 4.1)
We first present a simple lower bound on the cost of any symmetric equilibrium. This lower bound will turn out to be tight in case is fixed and is large. Next, we use this lower bound together with Proposition 4.6 to get a tight upper bound for small values of .
4.2.1 A Simple Lower Bound
We next prove that the cost of any symmetric Nash equilibrium is at least . This shows that the social cost of any symmetric equilibrium is at least twice the cost of the asymmetric optimum (in which the agents enter sequentially).
For any and any natural numbers it holds that in every symmetric equilibrium . In particular, .
Proof: We prove the claim by induction on n=. If the claim trivially holds. Assume that the claim holds for any such that , we prove the claim for any such that .
which proves the claim.
An immediate Corollary from the claim about is that the social cost of any symmetric equilibrium is at least :
For any and any in every symmetric equilibrium the social cost is at least , which is twice the cost of the asymmetric optimum.
From the lower bound on the cost of any symmetric equilibrium we can derive a lower bound on the players’ entrance probabilities.
For any and any it holds that in every symmetric equilibrium . Also, for every and it holds that .
Proof: Note that the cost of entering with probability is always greater than or equal to the cost the player exhibits, thus it holds that: . Now, by Claim 4.7 it holds that we get that
To complete the proof observe that the lower bound on is greater than when . In particular, it is positive when and thus .
4.2.2 An Upper Bound for Small
In this section we will use Proposition 4.6 with to show that .
For every and for every in every symmetric equilibrium:
Proof: We prove by induction over that for every and such that , it holds that . For the base it is easy to see that and . For the induction step, we assume the induction holds for every such that and and prove it holds for every such that and .
It is easy to see that is a nice candidate upper bound function. Hence, we can use Proposition 4.6 to get that:
For every such that and , it holds that .
Claim 4.11 below shows that for every , and every symmetric equilibrium, the next inequality holds.
Combining these two claims we get that as needed. Thus, we get that for every and for every for every symmetric equilibrium :
To complete the proof we note that for , and using standard arguments on the sum of a harmonic series we get that:
We now prove that :
For , every , and every symmetric equilibrium:
Where the rightmost inequality follows from the fact that by Taylor expansion and hence . It is easy to see that:
and the claim follows.
From Proposition 4.10 we derive the following corollary which clearly holds when .
For every fixed and every there exists such that for every we have that for every symmetric equilibrium it holds that and thus the social cost satisfies .
4.3 Bounds for Large (for Theorem 4.3)
4.3.1 Upper Bounds for Large
The upper bound on the cost of symmetric equilibria that we prove in Proposition 4.10 grows linearly in and logarithmically in . In this section we prove two upper bounds that grow much slower as a function of , only as , in the expense of a larger growth in . Both our bounds hold for any . The first one holds for any and gets a coefficient of on the term, while the second bound get a coefficient that converges to as grows, this rate of growth in is asymptotically tight, as we prove a matching lower bound (see Corollary 4.17).
For every :
For every and every symmetric equilibrium : .
For every , there exists such that for any , and every symmetric equilibrium :
Recall that , thus, The proposition implies the required upper bound for Theorem 4.3.
Proof: We prove both bounds by induction using Proposition 4.6. We use the upper bound function for the second statement and for the first statement. It is easy to see that is a nice candidate upper bound function according to Definition 4.5 and thus we will be able to use Proposition 4.6 as part of our induction.
In particular, we prove by induction over that
For every and every symmetric equilibrium , for every and such that , it holds that .
For every , there exists such that for any , and every symmetric equilibrium : for every and such that , it holds that .
It is easy to see that the base case holds for both statements. This is because for any and any we have that , . For the induction step, we assume the induction holds for every such that and and prove it holds for every such that and .
As is a nice candidate upper bound function, we can use Proposition 4.6 to get that: For every , there exists (in particular ) such that for any
For every , , such that , it holds that .
Thus, to complete the induction’s proof we use the statement that