Stiffing the tip as groups get larger ... like the prisoner's dilemma but ...? Is there a game theory analysis or other research or modeling describing the commonly perceived phenomenon whereby the amount by which people underpay a gratuity seems to be proportional to the number of people at the table?

In other words is there term of art for this behavioral pattern just as we use the phrase "Prisoner's Dilemma" to describe the game theoretical mechanics of two parties who can collude (to their mutual benefit) with risk, or betray their colleague in an effort reduce or eliminate their own risk?

(I would suggest that the mechanisms by which people exploit common resources and by which people fail to invest and maintain such resources (for example ecologically) are similar to this other group dynamic but I'm curious what research has been done on this).
Bystander Effect

The Bystander Effect could be your answer. The more people, the less personal responsibility. 


Person gets assaulted and mugged on a crowded street in broad daylight, nobody calls the cops. 
Software bugs slow company productivity almost to a halt, many employees, nobody reports the problem. 




Tragedy of the Commons

The tragedy of the commons has been studied from many perspectives. You can use that commons dilemma link as a jumping off point to hopefully happen across something relevant. But I would start by looking at the psychology behind the bystander effect.