Implementing the core of a two-person pure allocation game without free disposal or integer games

A class of Nash implementation problems of particular economic interest is that of implementing the core in games of pure allocation, or ‘cake division’. Mechanisms that have been proposed to solve this type of problem often involve off-equilibrium ‘punishments’ that lie in the interior of the feasible set; that is, in which not all the cake is allocated. They also tend to make use of extraneous devices such as ‘integer games’. Moreover, many mechanisms rely on there being three or more players. This note looks at the problem of implementing the core in a two-person ‘Edgeworth Box’ economy without free disposal and without integer games. The mechanism we propose, in or out of equilibrium, only involves allocations that lie inside the box. Agents’ strategy sets are limited to proposing an allocation and announcing a preference propfile. Free disposal can be viewed as allowing for a dummy player to act as a ‘sink’ for surplus, hence our restriction might be more in the spirit of a genuine two player problem. Moreover, free disposal may be an inappropriate assumption in some contexts, regardless of the number of players. For example, consider the allocation of hazardous waste. Here ‘free disposal’ is the threat of generating more waste and may mechanism that threatens this may not be ‘credible’ or ‘renegotiation proof’. The idea behind the ‘punishment set’ used by our mechanism in place of free disposal has been generalised by Moore and Repullo (1990) and by Dutta and Sen (1991). The implementation games that they suggest, however, use ‘integer games’. The main argument against the use of ‘integer games’ is that they add complexity to the agents’ problem. Not only are they somewhat extraneous, but they also result in unbounded strategy sets. Jackson and Palfrey (1990) and Sjostrom (1990) have looked at implementation in bounded mechanisms in a general context, but they allow ‘free disposal’. Moreover, both these papers