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.
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).
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 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).
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).
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.
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.