Figure 1. Payoff Matrix of Typical Prisoner s Dilemma This matrix represents the choices presented to the prisoners and the outcomes that come as the

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1 Proposal and Verification of Method to Prioritize the Sites for Traffic Safety Prevention Measure Based on Fatal Accident Risk Sungwon LEE a a,b Chief Research Director, The Korea Transport Institute, 370 Sicheon-Daero, Sejong-City, Korea a swlee@koti.re.kr Abstract: Under the assumption that the society is made up of selfish individuals, the increase in social utility can be achieved by guiding the individual selfishness to be aligned with the common good rather than naively publicizing the importance of following the traffic laws or using public transportation. In this paper traffic congestion issue is explained through game theoretic approach and getting to the equilibrium point is also explained. Policy measures are regarded as altering the utility functions of transport game and their consequences are analyzed in order to get to social optimum. Keywords: Game theory, Nash equilibrium, Stable equilibrium, Social welfare 1. APPLICATION OF PRISONER S DILEMMA TO TRANSPORT ISSUES Among the most rudimentary and renowned cases in Game Theory is the Prisoner s Dilemma. Its explanations are as follows. Two criminals are caught by the police. These two are accomplices. Due to lack of evidence, the police must depend solely on their confession in order to indict them. As the result, the police must separate the prisoners into different cells and offer them the following: the first to confess is immediately set free while one not confessing will be sentenced to 15 years in prison. In the case that both prisoners confess, they plead for extenuation under circumstances and will be sentenced to 2 years in prison. With no evidence and no confession, however, they will be let go after 2 weeks in detention. If it is given that these two can never communicate and will only be given short time to consider before making the decision to confess or not, what would be the result? The most probable outcome is that both confess and will be sentenced to 2 years in prison. This is because regardless of the decision of the accomplice, it is favorable for the prisoner to confess. To examine this more methodically, consider the following Payoff matrix. 309

2 Figure 1. Payoff Matrix of Typical Prisoner s Dilemma This matrix represents the choices presented to the prisoners and the outcomes that come as the results of their decision. The first sections of each cell show the given outcome of prisoner A, while the second show that of prisoner B. Let us consider the decision from prisoner A s perspective. Prisoner B has the option to confess or remain silent and so does prisoner A. In the case that prisoner B confesses, is it preferable for prisoner A to confess as well. Otherwise, prisoner A would be sentenced to 15 years in prison. Even in the case that prisoner B remains silent, it is more beneficial for prisoner A to confess since, despite the fact that the accomplice will be sentenced, prisoner A will be immediately let go. Therefore, whatever the choice that the accomplice makes, confessing will always be the optimal choice for prisoner A. Same goes for prisoner B as well. As the result, the decision to confess has become the stable equilibrium in which there is no incentive for the prisoners to break free from the equilibrium. Such stable equilibrium is not the most optimal among the possible outcome that the prisoners could reach. The optimal outcome of getting off with 2 weeks detention by mutually not confessing can never be established as the stable equilibrium. One of the concepts that is utilized in Economics is Pareto optimality: the state in which one s utility cannot be raised higher without reducing another one s utility. Pareto optimality, therefore, is a concept that emphasize purely on economic efficiency without consideration for equity. The stable equilibrium in the previous case is the only outcome that is not Pareto optimal while the rest can be said to be Pareto optimal. The stable equilibrium can therefore be considered by the 2 prisoners to be the most inefficient state. Despite the potential to reach a Pareto optimal outcome, why does the 2 prisoners inevitably reach a non Pareto optimal outcome? This results not from the prisoner s indiscretion but rather from rational decision making under the condition of mutual distrust. Each prisoner s strategy can be seen in diverse consideration as individually rational. Such strategy is not only 310

3 the dominating strategy of taking the preferable option regardless of the other side, but it is at the same time the prudent or minimax strategy of taking the preferable option in the worst case scenario. The end result shows, however, that the two prisoners were unable to achieve more desirable outcome. This non Pareto optimal result comes from two different reasons. First is the mutual mistrust. Due to the high possibility that the good faith for the benefit of all will most likely go unrewarded and the high potential risk, it becomes extremely difficult to expect loyalty, especially among accomplices. The second is the structure of the game. It is designed so that the independent and selfish rationality cannot help but to go against the collective interest. With the organized crime, reprisal towards the traitor is the common method of eliminating the inefficiency that comes with the said individual motivation among the members of the organization. In the case that the reprisal follows through, the payoff matrix is completely changed. Figure 2. Altered Prisoner s Dilemma According to the new payoff matrix, the optimal choice, regardless of the other s decision, is to not confess, resulting in the outcome that is the most favorable by the both prisoners. 2. GAME THEORY S TAKE ON TRAFFIC ISSUE Prisoner s dilemma represents the socioeconomic phenomenon of how individual rationalization leads to collective inefficiency. In terms of the transport sector, the radical increase in the number of cars in metropolis, along with increased pollution and wasted time and energy, is derived from such phenomenon. 311

4 Let us begin with the assumption that people have a choice between taking their household car and taking the available public transport such as subway, train, and bus. And t, 0<=t<=1, represents the proportion of the people who chose to take the public transport. a(t) and b(t) each represents the utility of individuals who could take the public transport or household car. a(t) and b(t) is represented in the following graph. Figure 2. Traffic Congestion Explained Game Theory This graph represents the following: if the number of people who take the public transport is greater than t1, the streets will be less crowded and utility of people who took the household car will be higher compare to that of people taking the public transport. On the other hand, if the number of people who take household car is greater than (1-t0), utility of people who take the public transport, such as the subway, is higher due to the traffic congestion on the streets. The equilibrium point in this choice of transportation is when [t0,t1] number of people choose to take the public transport and the rest take the household cars. In this case, the utility of both the people taking the public transport and the people taking cars will be the same. Such equilibrium can at the same time be seen as the stable equilibrium. For the purpose of demonstration, let us consider what would happen if a fraction of people changed their mind to take the public transport instead of cars. (=fraction) If enough people change their mind so that b(t)=a(t)<a(t+), the utility of people on the public transport will increase as long as the other people does not react to this change. However, this change to a(t+)<b(t+) will bring greater increase to utility of people taking cars and will restore a fraction of people to taking cars again. Likewise, when a fraction of people decide to take cars over the public transport for some reason, the equation will become b(t-)<a(t-), meaning that the utility of taking public transport will become greater which will entice a fraction of people back to taking public transport. It is extremely difficult to assess the changes in utility of each member in relations to the society to determine the universal utility of society as a whole. If the utility of the society is simplified as the sum of utility of its 312

5 members, ignoring the changes in economic utility amongst themselves, the above equilibrium can be said to be socially inefficient compared to any non-equilibrium points when t1 (t1 >t1) number of people take the public transport. Of course under such condition the point of achieving the highest social efficiency is when everyone takes the public transport, giving up the comfort of cars but getting to their destination fast and cheap. But such point can never be reached in the previous situation. If everyone uses the public transport, the calculating few will abuse the system to get greater utility of driving cars until the equilibrium is met and there is no more utility to be gained. Let us, for the sake of verification, consider an extreme case. Both a(t) and b(t) are increasing function for the variable t but a(t)<b(t). Figure 4. Altered Transport Game with Appropriate Policy Measures This case corresponds perfectly to the previously discussed Prisoner s dilemma. The only equilibrium to be established is the point where everyone suffers severe traffic congestion and inefficiency from everyone using their cars. This is the state in which the choices are made rationally at individual level but inefficient on the society as a whole. It is also the stable equilibrium from which the people do not have any incentive to move away. The state of highest efficiency in this case is again when everyone uses the public transport, achieving the maximum social utility at point a(1). In reality, such point will never become the equilibrium. Instead, the social utility will be b(0) where people suffer delays and confusion from everyone using cars. As for our country, the recent dramatic increase in number of cars and its usage signifies that the above equilibrium is rapidly moving left. This can be seen as the result of shift down along the a(t) curve due relatively poor quality of public transport or shift up along the b(t) curve due to increase in national income and the lower utilization cost of cars. These two 313

6 factors have likely influenced a complex reaction to bring about such outcome. As we have seen above, increase of public transport usage, with reduced cost and time, can bring social utility, but cannot be expected to be achieved without any effort. As in the case of organized crime, modifying the payoff structure is crucial to guiding the rational decisions made by individuals to achieve greater social utility. With the transportation case, the upward movement of the a(t) curve or the downward movement of the b(t) curve is the only solution to shifting the equilibrium to the right, signifying greater utilization of public transport. The upward movement of a(t), which represents the utility of public transport users, depends on the following: - Increase in quality of public transport - Decrease in price of public transport - Increase in speed (ex: private lane for buses) The downward movement of b(t), which represents the utility of car users, depend on the following: - Increase in cost of maintenance (ex: cost of fuel, tax rate) - Increase in parking fee/illegal parking control - Increase in passage toll As can be observed from the new graph, if both policies of improving public transport and penalizing car usage are properly enforced, the equilibrium point will move from t0 to t0, increasing the social utility from a(t0)=b(t0) to a(t0 )=b(t0 ). In countless other ways, analysis through game theory can be used to understand numerous traffic related phenomenon where individual s selfish rationalization leads to social inefficiency such as confusion in the intersection, delays in highways, etc. Of course for these cases there exists laws to penalize the unlawful. Ironically, however, lawlessness among the streets during the rush hour, due to its potential to become exacerbated if laws are enforced, often go unpunished. Instead it is more common to see roads less crowded to become the target for regulation. In such a case, the enactment of the traffic laws relies on a level of moral. As long as the above mentioned selfishness is established as the foundation, it remains extremely difficult to hope for stability that comes through traffic regulations. Therefore, despite the danger of increasing short term traffic congestion, it must be established that the violations of traffic laws will be strictly penalized, greatly reducing the number of such violation. 314

7 3. CONCLUDING REMARKS Under the assumption that the society is made up of selfish individuals, the increase in social utility can be achieved by guiding the individual selfishness to be aligned with the common good rather than naively publicizing the importance of following the traffic laws or using public transportation. Engineering new traffic regulations and enforcing them is crucial in influencing the citizens to behave in socially acceptable ways. This is one of the most important responsibilities of the government. But those in the government with the legislative power might have a hard time aligning these with his self-interest. In fact, it is often expected of the person to choose complacency which often will be the personally optimal choice. Unlike the rest of the citizens, however, the policy makers in the government is expected to uphold to a higher sense of morality, always valuing the public good over their self-interest. With these efforts, society can be expected to benefit from pragmatic regulations and increased welfare. REFERENCES John F. Nash Jr., Equilibrium Points in n-person Games, Mathematics 36: pp (1950). Yaron Hollander and Joseph N. Prashker, The Applicability of Non-Cooperative Game Theory in Transport Analysis, Transportation 33: pp (2006). 315