**
A Game-Theoretic Exploration of the Bystander Effect
**

Niene Tempelman, Hester van der Weij, & Evelien van Meeteren

ECB210: Applied Game Theory

Word Count: 2645

ECB210: Applied Game Theory

Word Count: 2645

**Abstract **

The murder of Catherine Genovese is the most well-known case of the bystander effect. The bystander effect suggests that the more bystanders witness an emergency, the smaller the chance that someone intervenes. This study will try to conceptualize the bystander effect by using game theory. By illustrating the cost of helping as the effect of the number of people who are present to a person’s payoff and thus to his strategy, this paper will try to explain the intricacies of the bystander effect. Furthermore, the two solutions most often mentioned in the literature, namely social rewards and punishments, are explored. The results indicate that when the number of people increases, the possibility of people helping decreases, however, this progress is significantly—and similarly—slowed by the introduction of rewards and punishments. Therefore, social rewards and punishments are possible solutions to the bystander effect that should be further explored.

The lack of help, in this case, can be explained by the bystander effect. The bystander effect suggests that the more bystanders witness an emergency, the smaller the chance that someone intervenes (Darley & Latané, 1968). In order to prevent situations where the bystander effect plays a part, solutions to the bystander effect should be tested and implemented. This paper will try to conceptualize the bystander effect by using game theory. This will be done by illustrating how the number of people present influences the cost of helping and the helpers payoff and thus his strategy. Therefore, our research question is:

*‘How can game theory explain the bystander effect and how can it give insights to possible solutions?’*

This paper will answer this question by first assessing the bystander effect in more detail, then explaining how a social psychological problem can be converted into a game-theoretic problem. Then the bystander effect will be examined through game-theoretic analysis to see if it is possible to mathematically explain this phenomenon. Furthermore, possible solutions of the bystander effect, such as rewards and punishments, will be investigated. I will finish by discussing the limitations and feasibility of using game theory to explain a complex social psychological phenomenon such as the bystander effect.

**Literature Review**

The bystander effect refers to the tendency of people to be less likely to help an individual in trouble when more people are present. This is explained by the principle of diffusion of responsibility, meaning that responsibility to help is divided and diluted between each witness, thereby decreasing the pressure for each individual to intervene (Thomas et al., 2016). Additionally, there are two other motives preventing people from helping. Firstly, there is “audience inhibition”, which is the fear that the situation transpires not to be an emergency, thus intervening would lead to embarrassment. Secondly, there is “social influence”, where witnesses screen the inaction of others and see this as cues to not intervene (Latané & Nida, 1981).

The predicament witnesses find themselves in can be explained as a social dilemma. Usually, a social dilemma is characterised by a dominant strategy and the resulting pure strategy Nash Equilibrium (NE), is a deficient equilibrium (Dawes, 1973). A dominant strategy is the strategy which is optimal for the player regardless of what the other player chooses. A NE is present when no player - after choosing their strategy - can enhance their expected payoff by altering their strategy when the other players do not change their strategy. A pure strategy NE occurs when the dominant strategy is chosen for sure, and not on a probability distribution. To overcome this social dilemma and attain a more efficient outcome, individuals must cooperate instead of playing their dominant strategy.

However, in the case of Catherine Genovese, there is no dominant strategy, resulting in two mixed strategy Nash equilibria, meaning a negative coordination where one of the players sacrifices themselves is needed. As Diekmann (1985) described, this is caused by the incentive for “free-riding” of someone else helping being bigger than the incentive to increase the collective good by helping the victim at the cost of the volunteer. However, if no one intervenes, all players lose. This is explained as the Volunteer’s Dilemma (VD), leading to the bystander effect.

As this social dilemma can be presented in a strategic way, game theory can help explain the tendency that when more people are present the probability of intervention decreases. Furthermore, game theory will help to give insights into the effects of introducing punishments and/or rewards into the payoff structure and if these changes result in increased incentive to overcome the bystander effect.

Rewards and punishments can be proposed as a solution for this social dilemma, as they are ways to make acting more attractive or not acting less attractive (van Lange, Rockenbach & Yamagishi, 2014). Although laws regarding people’s duty to help in life-threatening situations do exist, for instance, in the Netherlands, usually these laws only apply in some rare cases and there are not many known cases of people actually being punished for their lack of action (Art. 450 Wetboek van Strafrecht, 1984). Rewards and punishments can be both tangible and intangible, meaning that a reward can, for example, be in the form of money or in the form of positive media attention. Similarly, a punishment could be a fine or negative media attention. Many researchers have claimed that intangible rewards and punishments can be effective in societies with strong social norms regarding helping others, which is the case in more collectivist countries such as China and Japan (Chekroun & Brauer, 2002).

**Game-Theoretic Analysis**

The elements of the game-theoretic model include players, actions, information structure, payoffs, and the rules of the game. In order to model the bystander effect, n players are needed. The actions for the player i include act or not act. For the player(s) n-i, the actions are either not acting, or at least one person acting. The letter n refers to the number of people playing the game. So if there are two players in the game, n-i is only one player, but if there are three players, n-i will be two players and so on. The information structure consists of imperfect information, as the strategy of n-i is not observable for i, and vice-versa. The payoffs for the players include the value of the life of the person in need (v) and the cost of acting (c), where v > c, because when no-one acts, person i prefers to act. When player i acts and player n-i does not act, the payoff will be v-c for player i and v for player(s) n-i. respectively, and when players i and n-i do not act, payoffs will be 0 for all players. As the game is symmetric, the payoffs are similar for players i and n-i. The rules of the game specify that it is a simultaneous, non-cooperative game that is played once with v-c > 0, meaning if no-one acts, i prefers to act, and v > v-c, meaning if at least one person acts, i prefers to not act.

**Figure 1**

*The strategic form of the game with n players*

*The same conclusion can be made with n = 2

In our game, there are two asymmetric pure-strategy NE, meaning that for each situation a player prefers a different action (illustrated by

**Figure 1**). If player i chooses to act, players n-i will choose not to act and when player i chooses not to act, players n-i will choose to act. Therefore, there is no dominant strategy as the optimal choice depends on the actions of the other player. This coordination problem makes it impossible to determine at which NE the players will end up. Therefore, a mixed strategy NE, which stipulates the probability that will explain which NE the players will find themselves, is proposed. A mixed strategy NE thus means that one player uses a random strategy and no player can raise their expected payoff by using an alternate strategy. In this case, the mixed strategy NE will entail the probability of person i choosing to act, where p is the probability of acting and 1-p probability of not acting (Diekmann, 1985). In our analysis, we will first theorise the game of a VD in a strategic form with n = 2 players to analyse the dynamics of the VD (see Appendix A). The probability of person 1 acting is p = (v-c)/v where p ∈ [0,1] (see Appendix B). This means that the higher the value of v (ceteris paribus), the higher the probability of acting. Furthermore, the higher the value of c (ceteris paribus), the lower the probability of acting. However, an increase in c has more effect on the probability of someone helping than the same increase in v. Thus, the incentive of helping depends for n = 2 more on the cost of helping than the value of someone’s life. Nevertheless, the n = 2 game does not depend on the number of bystanders, and is thus not representative of the bystander effect, as this game only depends on the values of v and c.

Therefore, we introduce a game with n-i players. The strategic form shows two NE for act and not act, and not act and at least one person acts (See

**Appendix C**). Therefore, the mixed strategy NE has to be constituted from the joint probabilities of all players. This will mean the joint probabilities (1-p)n-1 for players n-i not acting and 1-(1-p)n-1 for at least one person of n-i acting (See

**Appendix D**). This allows for the following mixed NE to be derived; the probability of a person i acting is p = 1-(c/v)1/n-1 where p ∈ [0,1] (see

**Appendix E**). The graph below illustrates the effect of the increase in n on different values for v and c (see

**Figure 2**and

**Appendix F**).

**Figure 2**

*Probability of Acting in Correlation to the Numbers of Players*

The graph shows that with an increase in n (the number of players), the probability of a person i acting decreases. The graph also shows that when the payoff of person i acting, namely v-c, decreases (from 3 to 2), the probability of person i acting decreases. Even with different values and ratios for c and v, the probability still decreases when there is an increase in n. Therefore, the bystander effect is clearly shown.

*
*The Effects of a Criminal Record*
*

As mentioned in the literature review, in the Netherlands there are laws punishing people with entries into their criminal record if they fail to act in an emergency situation. Therefore, the payoff structure is altered by adding the consequences of a criminal record r when a person does not act, where r<c. Although the pure strategy NE stays similar (See *The Effects of a Criminal Record*

**Appendix G**), the mixed strategy NE changes. The mixed strategy NE proposes that the probability of a person i acting is p = 1-((c-r)/v)1/n-1 where p∈[0,1] (see

**Appendix H**). The graph below depicts the influence of a criminal record on the probability of helping (see

**Figure 3**).

**Figure 3**

*Probability of Acting in Correlation to the Numbers of Players with the Cost of a Criminal Record*

Unfortunately, the graph depicts that even though a punishment increases the probability of someone acting, it does not cancel out the bystander effect (see

**Appendix I**).

**
The Effects of a Reward
**

As mentioned in the literature review, a reward can also be offered if a person acts in an emergency situation. Therefore, the payoff structure is altered by adding the benefits of a reward b when a person does act, where b<c. Again, the pure strategy NE stays similar (see **Appendix J**), but the mixed strategy NE changes. The mixed strategy NE proposes that the probability of a person i acting is p=1-((c-b)/v)1/n-1 where p∈[0,1] (see

**Appendix K**). The graph below depicts the influence of rewards on the probability of helping (see

**Figure 4**).

**Figure 4**

*Probability of Acting in Correlation to the Numbers of Players with the Benefit of a Reward*

Unfortunately, the graph depicts that even though a reward increases the probability of someone acting, it does not prevent the bystander effect (see

**Appendix L**). Although the payoffs of the punishment and rewards are placed on not acting and acting respectively, the mixed NE stays the same. This leads to the conclusion that the effects of the bystander effect are indifferent between rewards and punishment. Additionally, a (increase in) punishment/reward still increases the probability of acting. Therefore, these laws should stay in place and research into other solutions to the bystander effect should be conducted.

Lastly, we will use our model to explain the real-life example of Catherine Genovese.

**Figure 5**

*Probability of Acting in Correlation to the 37 Witnesses of the Catherine Genovese Case*

The graph shows clearly that when there are 37 people witnessing an emergency, the probability of someone acting approaches 0 (see

**Figure 5**). Even if punishments and rewards had been in place to prevent the bystander effect, the probability of someone acting as predicted by our model approaches 0.

**
Conclusion**

Through the use of game theory, this paper has analysed the bystander effect and has given insights on the choice situations among the players. It has provided a mathematical understanding to complement the psychological side on why people are less inclined to help others when more people are present. Nonetheless, this research is not without limitations. By using the standardized n for the number of players, individual differences, such as different tendencies towards altruistic behaviour, are disregarded. These differences can either be because of inherited characteristics or because of cultural differences, e.g. growing up in an individualistic or collectivistic society. For example, rewards and punishments will have a bigger effect on reducing the bystander effect in collectivistic societies compared to individualistic societies (Leung & Bond, 1984). Further research on the bystander effect should therefore focus on differences in altruism, by either creating a game in which chance determines whether a bystander is altruistic or not altruistic and solve this VD with the Bayesian NE solution concept, or by letting the researchers choose the number of altruistic players.

Additionally, since this paper is neither an economic nor a criminological paper, it is limited in assigning feasible values to the rewards and punishment introduced in the game, making it difficult to apply the formula to real-life situations. For example, what constitutes a punishment and what value do we give that respective punishment? Furthermore, the probability for rewards and punishments were based on b < c and r < c respectively. However, this meant that it was impossible to look at the effects of b and r being higher than the cost of helping. As the mixed strategy NE is not suited for this problem, further research should use other solution concepts. Another controversial but noteworthy point is that the bystander effect, and explicitly the case of Catherine Genovese, are according to some portrayed much more dramatic than it actually was. Catherine’s brother claimed that some witnesses did try to help her and that there were not 37 eyewitnesses but only six, of which only two people saw the attack (Jancelewicz, 2016). This perspective indicates that more research on the concept of the bystander effect itself is needed. Modelling game theory cannot always predict real-life situations, since it includes many simplifications.