# The Vote with the Wallet Game: Responsible Consumerism as a Multiplayer Prisoner's Dilemma
## 4 The Power of Coordination
### The Coalition of the Willing and the Paradox of Altruism  
Here we provide a more intuitive example illustrating how a coalition can work. Suppose the existence of a share \(\pi^{*}\) of non-income constrained individuals with other-regarding preferences of the form \(a^{*}\) where \(a^{*}>c-\pi^{*}b\) or \(\pi^{*}>\frac{c-a^{*}}{b}\). These individuals would nonetheless vote with the wallet for the responsible product if they could coordinate and form a _coalition of the willing_ which synchronises their voting choices. Based on the above inequality, \(\pi_{min}=\frac{c-a^{*}}{b}\) can be defined as the minimum responsible vote with the wallet coalition threshold required by individuals with other-regarding preferences higher or equal to \(a^{*}\). Figure 4 shows the three-dimensional \((\pi,a^{*},c)\) threshold of feasible parameters for a 'coalition of the willing' when conveniently normalising \(b=1\). The plane \(c=0\) always belongs to our set because it corresponds to the situation in which the cost of responsible voting is null, and therefore the benefit for each individual of the coalition satisfies \(b+a^{*}\geq b\) for all \(a^{*}\). Hence, when \(c=0\), we have \(\pi_{min}=0\). However, when \(c\) increases, the other-regarding preference coefficient \(a^{*}\) also needs to increase in order to keep \(\pi_{min}=0\) (segment \(\overline{OC}\)). This may be interpreted as the higher the cost of voting responsibly, the higher the other-regarding preferences of individuals to ensure compatibility with the lowest coalition threshold. On the other hand, the coalition of the willing must be larger with a higher cost \(c\) in order to convince individuals with other-regarding preferences to vote responsibly. (As an extreme case, when the cost is too high and other-regarding preferences are too small, \(\pi_{min}=1\), implying that there will always be (at least one) individual(s) with buying the conventional product as a best strategy (segment \(\overline{AB}\)). Note as well that when all players are willing to vote responsibly and every individual has a preference to vote responsibly equal to the benefit of the public good component (the numeraire \(b\)), then the extra cost can be up to twice as much the numeraire \(b\) (point \(B\)).). Suppose now that an organisation can form a coalition of the willing so that a share \(\pi^{*}>\pi_{min}\) (this is a necessary condition to make responsible voting with the wallet strategy nonetheless incentive compatible) of altruists reveal their strategy \(p_{t_{1}}=(1,1,1,1)\) at \(t_{1}\) (regardless the outcome at \(t_{0}\)). For instance, he strategy can be revealed through a cash mob (Cash mobs are media (video recorded) events where an organised group of sellers gather into a retail outlet to buy a given product and intend to communicate its decision to the general public. For a reference on the US cash mobs see [[http://cash-mobs.com/](http://cash-mobs.com/)]([http://cash-mobs.com/](http://cash-mobs.com/)).) where the coalition plays the responsible voting with the wallet strategy and announces her strategy for the future. After the strategy is revealed the remaining players will however vote for standard product if \(c<(1-\pi)b+a^{*}\) or because they will fall into the PD if  
\[\frac{1}{n}b+a^{*}<c<(1-\pi)b+a^{*}\]  
Assuming that the above inequality holds, in order to avoid the PD the coalition of the willing needs t announce addedits strategy at \(t_{1}\) as \(p_{t_{1}}=(1,0,1,0)\), which consists in punishing the free riders at period \(t_{1}\) by not voting with the wallet responsibly. Given the coalition of the willing's strategy, the benefit in \(t_{1}\) for the myopic self-interested individuals who buy the conventional product is  
\[\pi^{*}b-(b+a^{*}-c)=c-a^{*}-(1-\pi^{*})b\]  
The potential loss for out-of-coalition individuals from the punishment occurring when members of the coalition of the willing deviate from their responsible voting strategy is  
\[\frac{\pi^{*}b}{1+\delta}\]where \(\delta\) is the discount rate measuring players' patience.