# The Vote with the Wallet Game: Responsible Consumerism as a Multiplayer Prisoner's Dilemma
## 2 The Simplest Model Representation: A 2-Player Static Game
Evidence from behavioural studies highlights that individuals have other-regarding elements in their preferences ranging from (positive and negative) reciprocity [30], inequity aversion [31; 32], other-regarding preferences [33], social welfare preferences [34], and various forms of pure and impure (warm glow) altruism [35; 36]. A meta study of [37] examines results from around 328 different Dictator game experiments for a total of 20,813 observations. The result is that only around 36 percent individuals follow [PERSON] rationality and give zero (based on these numbers the author can reject the null hypothesis that the dictator amount of giving is 0 with \(z=35.44\), \(p<0.00001\)) and more than half give no less than 20 percent.) The third factor (\(c\)) (We assume \(c\in[0,+\infty)\) since it represents an extra-cost between a standard products and a SR product, when the former is cheaper than the latter. Alternatively, it could be assumed that \(c\in(-\infty,+\infty)\). However a negative \(c\) makes the problem trivial. Our theoretical analysis hence applies to the more frequent and reasonable cases in which CSR adds extra costs.) measures the cost of voting with the wallet, namely the extra cost, if any, paid by the consumer when choosing a product of a responsible company vis-a-vis a product of comparable quality and lower price of another company which falls below the responsibility standards of the former. We as well assume for simplicity that \(Y_{i}>c\) for all \(i=1,2\) (where \(Y_{i}\) is the income of player \(i\)), that is, all players' decisions to vote or not for SR products depend only on utility considerations and are not constrained by lack of income. The above described game can be represented by \(G=(N,(S^{i})_{i\in N},(U_{i})_{i\in N})\), where \(N=\{1,2\}\) is the set of players, \((S^{i})_{i\in N}\) is the set of actions, and \((U_{i})_{i\in N}\) is the set of payoffs described in (1). The payoff matrix writes \begin{tabular}{c c|c|c} \multicolumn{3}{c}{Player 1} & \multicolumn{3}{c}{\(vR\)} \\ \cline{2-3} \cline{5-5} \multicolumn{1}{c|}{\(vR\)} & \multicolumn{1}{c|}{\(b+a-c,b+a-c\)} & \(\frac{1}{2}b+a-c,\frac{1}{2}b\) \\ \cline{2-3} \cline{5-5} \multicolumn{1}{c|}{\(vS\)} & \multicolumn{1}{c|}{\(\frac{1}{2}b,\frac{1}{2}b+a-c\)} & \(0,0\) \\ \cline{2-3} \cline{5-5} \end{tabular}  The game \(G\) has always a unique NE, which is \((vS,vS)\) if \(\frac{1}{2}b+a<c\) and \((vR,vR)\) otherwise. The parametric conditions creating a PD in the game are  
\[\frac{1}{2}b+a-c<0<b+a-c<\frac{1}{2}b\]  
or  
\[\frac{1}{2}b+a<c<b+a \tag{2}\]  
That is, when (2) holds, the (unique) NE \((vS,vS)\) is Pareto dominated by the strategy pair \((vR,vR)\) which yields the highest payoff for both players. By considering \(b\) and \(a\) as product and individual characteristics respectively, and \(c\) as a parameter which may differ depending on the characteristics of the market, the three regions of equilibria generated by different values of the cost of voting responsibly are illustrated in Figure 1.  
More specifically, given (2), we are not anymore in the PD area when the cost of voting responsibly is too high (\(c\geq b+a\)) or too low (\(c<\frac{1}{2}b+a\)), which could be the case of products with a relatively low (or zero) cost of voting for socially responsibility (see our discussion in Section 2.2 which follows). Please note that if \(c=b+a\), the NE are \((vR,vR)\) and \((vS,vR)\), and both are inefficient, while if \(c=\frac{1}{2}b+a\), we have that \((vR,vR)\) is efficient and \((vS,vS)\) is inefficient. Based on the above described features the originality of the game in the PD literature lies in its 'hybrid' provision-PD game characteristics ([38] classify PDs into four categories (provision, commons, altruism, selfish) according to the private/public benefits and costs to players and to the action/inaction choices related to the 'cooperation' and 'defection' strategies.) where both classical 'cooperation' and 'defection' strategies require an action.