I Introduction
Instead of classical yet inefficient traffic lighting systems, a FirstComeFirstServe (FCFS) based autonomous intersection control, which utilizes the connectivity of the autonomous agents with each other and the infrastructure, e.g., an intersection manager, has been introduced in [1]. Each agent requests the usage of the intersection at a certain time slot, he/she has desired, and the intersection manager confirms the request if it is available, i.e., not has already been allocated for other agents, or proposes a counter request. This simple approach improves the efficiency of the intersections substantially in terms of travel time of the agents. However, the requests of the agents are evaluated one by one. In order to increase the efficiency further, Reference [2] proposes a combinatorial auction based approach, in which the intersection is allocated to the agent who values the most, instead of the agent who has requested the earliest. This approach leads to relatively complex algorithms to compute the intersection allocation and requires a payment system. In [3], the authors have proposed a chickengame [4] inspired intersection control. The proposed game includes two players, where the players aim to minimize their delay while also avoiding any collision; and the intersection manager controls their actions, i.e., swerve or not, to achieve a Nash equilibrium [5] of the game. Recently, in [6], the authors have proposed an information driven intersection management mechanism, which improves the quality of transportation by prioritizing certain vehicles based on the information reported by them and ensures truthful disclosure of that information via a payment mechanism.
In this paper, we introduce an intersection game formulation. Different from the widely known chicken game [4], here, we consider an intersection control scenario, where two noncooperative autonomous agents (drivers) seek to use a single intersection resource at a specific
time they desire. In that respect, a desired passing time for an agent can be considered as an estimated time such that he/she can pass through the intersection with minimum loss of comfort, e.g., without acceleration or deceleration (excluding the deceleration necessary to pass through the intersection safely). However, there is a certain time that the agents will need while using the intersection resource. Therefore, there can be a conflict between their desired intersection usage.
In order to avoid conflicts which might lead to accidents, the intersection manager provides a FCFS based resource allocation protocol so that the agent who desires to pass through the intersection earlier will pass earlier, and if both desire to pass at the same time, then both have the same chance to pass first. To this end, the agents report their desired passing times to the intersection manager and the manager allocates the resources accordingly [1]. We note that, here, in order to increase the efficiency of the intersection usage, the request of two agents are evaluated together, which is different from the proposed approach in [1], where the agents’ requests are evaluated one by one. However, while the evaluations of more than one request can increase the efficiency, in a noncooperative environment, the agents can also report strategically, e.g., they may not reveal their true desired passing times, in order to minimize the deviation of the times that the manager allows them to use the intersection from their desired passing times. Hence, we aim to formulate the strategic behavior of the agents in this noncooperative environment. Furthermore, instead of any Nash equilibrium as in [3], we seek to select the socially optimal one among multiple equilibria, while designing the intersection control mechanism.
The main contributions of the paper are as follows:

We model the strategic behavior of the noncooperative agents in FCFS based intersections and formulate the corresponding equilibrium points analytically.

We provide socially optimal intersection usage allocations and compute the socially optimal equilibrium strategies for the agents.

We propose a reliable intersection control mechanism, which ensures that the strategic agents reveal their private information truthfully, i.e., cannot rig the intersection control mechanism by misreporting their private information strategically.
The rest of this paper is organized as follows: In Section II, we formulate the problem for two strategic autonomous agents and a single intersection resource. In Section III, we analyze the equilibrium scenarios in a strategic environment. We provide a strategyproof intersection mechanism, where the agents reveal their private information truthfully, in Section IV. We conclude the paper with several remarks in Section V. Appendices include proofs of technical results.
Ii Problem Formulation
Consider two noncooperative autonomous agents, agent and agent, seeking to pass through a single intersection, which is equipped with a roadside unit that has communication radios. In close proximity of the intersection, agents, equipped with communication radios, can request the temporal intersection usage from the roadside unit, which reserves the intersection temporally to the agents according to a FCFS protocol and certain safety constraints. As an example of safety constraints, the roadside unit considers the necessary time for the agents to pass through the intersection while scheduling the intersection usage.
Each agent, for , has access to private information: , denoting the earliest possible and the desired passing times through the intersection, respectively, where is a totally ordered finite set such that for any consecutive elements we have and for all , , where and are specified upper and lower bounds, respectively, with, e.g., . can be viewed as the highest precision for time reporting. Each agent needs certain amount of time while passing through the intersection, which is denoted by , and we assume that .
Agents report their passing times and the manager controls the intersection usage according to FCFS protocol. In particular, if agents report , then the allocated times would be given by , since agent comes to intersection first, and . Furthermore, in case of equality, i.e., , the manager allocates the intersection to one of them first randomly, e.g., by flipping a fair coin, for the sake of fairness. Therefore, for given reported times , the protocol computes the corresponding allocated times, i.e., , uniquely.
Note that if the agent, who has the right to pass first, starts to use the resource after his/her allocated time, the time he/she will need to pass through the intersection might violate the allocated time for the other agent. Or if that agent does not use the intersection before the time allocated for the other agent, then the agent who has the right to pass second cannot start to use the intersection at his/her allocated time due to a security precaution. Therefore, without certain regulations, the agents would reveal practically unhelpful information. As an example, in the case of conflicting interests, if the agent with later desired passing time reports that his/her desired passing time is just now, then the manager would allocate the intersection to that agent to use starting from now unless the other agent has also reported to pass now. In order to avoid such cases, we consider that there is a regulation that the agents must use the intersection exactly at the allocated time. Since the allocated times are computed based on the reported desired passing times, it is the agents’ responsibility to ensure that they will pass through the intersection as early as their reported times. However, even though this regulation can prevent the aforementioned cases, as we show later, this is not sufficient to incentivize the agents to reveal their desired passing times truthfully.
We consider that the agents are identical and have payoff functions penalizing the deviation of the allocated times from the desired passing times :
(1) 
where is a strictly increasing, strictly convex function on , e.g., . We consider Nash equilibrium [5], in which agents do not have any incentive to change their actions unilaterally, and an agent has an incentive to change his/her action if he/she can have a reduced payoff. Therefore, an action pair leads to an equilibrium provided that
where . In the next section, we examine these equilibrium scenarios in detail.
Iii Equilibrium Scenarios in a Strategic Environment
Without loss of generality, we can consider that is earlier than or equal to , i.e., . In such cases, if both agents reveal their desired times truthfully and , agent would pass through the intersection first at his/her desired time and agent would pass at the time , and if then both agents would have equal chances to pass through the intersection first. However, the agents are strategic and a strategic agent would reveal the information truthfully if it is the best strategy according to his/her objective, e.g., minimizing the payoff function (1).
We point out that the agents do not have conflicting interests if . Hence in such cases, the agents would report their desired passing times truthfully and the manager would be able to allocate the intersection exactly at their reported times. Otherwise the agents have conflicting interests and in those cases, truthfulness is not an equilibrium achieving strategy for the agents in general. In that respect, the following cases lead to interesting equilibrium scenarios, e.g., even though , agent can incentivize agent to report an earlier passing time than and under certain conditions, the intersection might even be allocated to agent first. Next, we examine these scenarios in detail.
Lemma 1. Let . Without loss of generality, suppose . Then, the action pairs lead to an equilibrium if there exists such that . If is empty, i.e., , then the action pair , where , leads to an equilibrium.
Proof. The proof is provided in Appendix A.
At equilibria of certain cases, interestingly, the reported time of the agents can even be less than their earliest passing time even if there is a regulation making sure that each agent must pass through the intersection at his/her allocated time. However, since the reported time can be different from the allocated time, even though that agent knows that he/she cannot pass through the intersection that early, by reporting in that way, he/she incentivizes the other agent to report a far earlier time such that his/her allocated time will not be the reported time. We also note that the deviation of this allocated time from the desired passing time can be less than the one when he/she has reported as early as the earliest passing time. Therefore, such equilibria can be more preferable for the agent that has a later desired passing time.
Lemma 2. Let and , then the actions pairs lead to an equilibrium if there exists such that . Additionally, the action pairs , where such that , also lead to an equilibrium.
Proof. The proof is provided in Appendix B.
Lemma 3. Let and , then the action pairs lead to an equilibrium if there exists such that . However, if is empty, then the action pair , where , leads to an equilibrium.
Proof. The proof is provided in Appendix C.
Lemma 4. Let and , then we have two different cases, where the first passing agent differs. If , the action pairs lead to an equilibrium if there exists such that . However, if is empty, then the action pair , where , leads to an equilibrium. On the contrary, if , the action pairs^{1}^{1}1Here, agent passes through the intersection first at the equilibria. lead to an equilibrium if there exists such that . However, if is empty, then the action pair , where , leads to an equilibrium.
Proof. The proof is provided in Appendix D.
Lemma 5. Let , then there exist multiple equilibria such that the equilibrium achieving action pairs are given by provided that there exists such that
(2) 
Otherwise, there exists a unique equilibrium, where and .
Proof. The proof is provided in Appendix E.
Based on Lemmas 15, we have the following proposition.
Proposition 1. There exists at least one pure equilibrium point in the twoagent intersection game.
Proof. As seen in Fig. 6, Lemmas 15 cover all the conflicting cases and at each case, there exists at least one pure equilibrium. Furthermore, if there are no conflicts, then truthfulness leads to pure unique equilibrium.
We note that in general there are multiple equilibria. However, in the case of multiple equilibria, certain equilibrium points are more preferable for certain agents. Additionally, since there are multiple equilibrium points, taken actions might not yield an equilibrium. In the following section, we design a mechanism to mitigate such issues.
Iv StrategyProof Intersection Mechanism
We seek to design an intersection mechanism between the agents and the intersection manager, where the strategic agents report their private information to the mechanism and the mechanism plays on their behalf, i.e., reports certain times to the intersection manager based on the agents’ reports. Note that the agents have selfish objectives, which is to minimize their own payoff functions. However, we seek to design a mechanism which leads to minimization of the social choice function:
(3) 
Note that our social objective (3), as a mechanism designer, can lead to different allocations from the FCFS protocol. Particularly, the socially optimal allocated times would be given by the following minimization problem:
(4) 
If there is no conflict of interest, i.e., , then the allocated times by the manager according to FCFS protocol also lead to the socially optimal ones. However, if there is a conflict of interest, since is strictly increasing and strictly convex function, the socially optimal allocated times are such that the deviations of the allocated times from the corresponding desired times are the same for each agent. In particular, we have^{2}^{2}2Note that we have assumed . if , , , where ; if and , , , where ; if and , , , where .
Remark 1. We emphasize that the socially optimal time allocations do not depend on the payoff functions directly due to the assumption that agents are identical and is a strictly increasing and strictly convex function. Therefore, as a mechanism designer, we do not need to know the exact payoff functions except the assumption about their structure.
Furthermore, in the cases of conflicting interests, the socially optimal allocated times cannot be achieved if both agents reveal their desired passing times truthfully since the manager cannot allocate a time before the reported times. If the agents behave strategically and not necessarily reveal the information truthfully, the socially optimal allocation may also not be achievable at any of the equilibria due to the earliest passing times of the agents. As an example, in Figs. 7 and 8, we demonstrate all the possible pair of time allocations, i.e., a point in the shaded areas of the graphs, for the cases on the top of each figure. Then, the socially optimal allocation is at and while the action pair , where leads to an equilibrium in Fig. 7 and the action pair , where leads to an equilibrium in Fig. 8. Therefore the corresponding time allocations at equilibria are given by and in Fig. 7 and given by and in Fig. 8. However, for the case in Fig. 7, there does not exist a such that and . Additionally, for the case in Fig. 8, there does not exist a such that and . Existence of such would imply that the socially optimal allocation is in the set of allocations at equilibria. Therefore the socially optimal time allocation may not lead to an equilibrium. However, in such a case, since the set of equilibrium points is finite, there exists at least one equilibrium that is socially more preferable than, or equally preferable with, the other equilibrium points. We call such an equilibrium “socially optimal equilibrium” and denote the corresponding time allocations by (e.g., see Figs. 7 and 8).
Let , and given , let denote the set of all the corresponding equilibrium points and be the corresponding set of allocated passing times. If there is a unique pure equilibrium, e.g., the case when and , then this is also the socially optimal equilibrium point. Otherwise, the socially optimal equilibrium leads to the allocations , which are given by
(5) 
The following theorem characterizes these allocations.
Theorem 1. Suppose that there are multiple pure equilibria, and yet the socially optimal allocations cannot be achieved at any of these equilibria. If , let and ; otherwise let and . Then, the allocations in the socially optimal equilibrium are given by
(6) 
where .
Proof. The proof is provided in Appendix F.
We reemphasize that also the time allocations in the socially optimal equilibrium, , do not depend on the payoff function directly. The following corollary provides the socially optimal equilibrium points such that .
Case  Condition  Assignment of  

No Conflict  
Lemma 1  
if  , where  
else if  
else if  
else  
Lemma 2  and  
if^{3}^{3}3Note that this is one of the essentially unique equilibria, which lead to the same time allocations. Furthermore, truthfulness of both agents leads to the socially optimal equilibrium.  
else if  
else if  
else  
Lemma 3  and  
if  , where  
else if  
else if  
else  
Former Case  and  
in Lemma 4  if  , where  
else if  
else if  
else  
Latter Case  and  
in Lemma 4  if  
else  
Lemma 5  
if  , where  
else if  
else if  
else 
Corollary 1. If , let and ; otherwise let and . Let be defined by . Then, the socially optimal equilibrium is given in Table I.
Proof. These results follow from Theorem 1 and Lemmas 15. We select the equilibrium point within such that the corresponding time allocation is closest to the socially optimal allocation.
Next, the following corollary provides a truthful mechanism for intersection control in strategic environments.
Corollary 2. Consider an intersection mechanism, which asks the agents to report both their earliest possible and desired passing times, , for ; and plays for them the corresponding socially optimal equilibrium actions, i.e., reports passing times to the intersection manager: , which are formulated in Corollary 1. Then, this intersection mechanism is a strategyproof mechanism, i.e., it incentivizes the selfish agents to reveal their private information truthfully.
Proof. This follows from the revelation principle [7], which implies that the socially optimal equilibrium can be implemented by an incentivecompatibledirectmechanism, in which both agents reveal their private information truthfully. The intersection mechanism already plays the best actions strategically for both agents, which lead to the corresponding socially optimal equilibrium; therefore the agents do not have any incentive to misreport their private information with the aim of reducing their own payoff function further.
V Conclusion
In this paper, we have studied the truthfulness of intersection control, when the intersection manager evaluates the requests of more than one agent, specifically two agents, in a noncooperative environment. We analyze equilibria for the agents’ actions such that at an equilibrium, given that the other agent has taken that action against his/her action, he/she has no incentive to change his/her action unilaterally. We have formulated all equilibria for all possible scenarios. We have shown that there always exists a pure equilibrium and there even exist multiple pure equilibria in general. We have then characterized the socially preferable equilibrium within all the equilibria with respect to a certain social objective. Finally, we have designed a strategyproof mechanism, where the agents cannot exploit the intersection control to get benefit in terms of their selfish objectives. We point out that the intelligent intersection control process in a noncooperative environment even with two agents requires a careful consideration of all the possible cases with respect to earliest and desired passing times of the agents, and the time that the agents need to pass through the intersection. By formulating the strategyproof mechanism, this paper has also provided a straightforward, yet nontrivial, guideline to extend to scenarios where there are more than two agents.
References

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K. Dresner and P. Stone, “A multiagent approach to autonomous intersection
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[3]
M. Elhenawy, A. A. Elbery, A. A. Hassan, and H. A. Rakha, “An intersection gametheorybased traffic control algorithm in a connected vehicle environment,” in
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Appendix A
Aa Proof of Lemma 1
Given that and , we have the following cases, which are partially demonstrated in Fig. 1: If both and are less than or equal to , there exists a unique pure equilibrium, where both agents report , and the intersection is allocated to one of the agents at
with equal probability and the other agent is allowed to pass through the intersection when
. The action pair leads to an equilibrium since if agent reports , then the best response of agent is to report , and given agent2 has reported , agent has no incentive to change his/her action. Furthermore, this is the only pure equilibrium. If and , then there exist multiple pure equilibria. In particular, if agent reports , where such that , then the best response of agent is to report and given that agent has reported that, agent has no incentive to change his/her action since agent cannot pass through the intersection before . Note that the set is not empty since . Finally, if , then there exist multiple pure equilibria. Correspondingly, the equilibrium achieving action pairs are given by , where such that . If there is no such , i.e., , then we have a unique pure equilibrium, where both agents report .AB Proof of Lemma 2
Given that and , we have the following cases, which are partially demonstrated in Fig. 2: If both and are less than or equal to , there exist multiple equilibria, where the equilibrium achieving action pairs are given by , where such that . Additionally, the action pair , where such that , also leads to an equilibrium. Note that if agent reports a time earlier than , e.g., , the best response of agent is to report a far earlier time, e.g., . However, given that agent has reported , now, agent has an incentive to change his/her action since by reporting , agent can pass through the intersection exactly at his/her desired passing time, which is the least possible payoff he/she can get. Therefore, such action pairs do not lead to an equilibrium. If , then there exist multiple equilibria, where the equilibrium achieving action pairs are given by , where such that . Correspondingly, the action pair , where such that , also leads to an equilibrium.
AC Proof of Lemma 3
Given that and , we have the following cases, which are partially demonstrated in Fig. 3: If and , then there exist multiple pure equilibria, where the equilibrium achieving action pairs are given by , where such that . Note that the set is not empty since . If , we have two cases. If , there exist multiple pure equilibria, where the equilibrium achieving action pairs are given by , where such that . Or if , there exists a unique pure equilibrium, where . If , there exist multiple pure equilibria, where the equilibrium achieving action pairs are given by , where such that . Note that the set
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