= \mathbf y  \right ]}{[ \allocs(\types) = \mathbf y ]}{[ \allocs(\types) = \mathbf y ]}{[ \allocs(\types) = \mathbf y ]} }}} = 1$.

Agents have quasi-linear utilities, that is, when reporting type $\typei'$, agent $i$’s utility is ${\operatorname{\mathbf E}_{}{{\mathchoice{ \left [ \langle \typei, \alloci(\typei', \typesmi) \rangle   \right ]}{[ \langle \typei, \alloci(\typei', \typesmi) \rangle  ]}{[ \langle \typei, \alloci(\typei', \typesmi) \rangle  ]}{[ \langle \typei, \alloci(\typei', \typesmi) \rangle  ]} }}} - \payi(\typei', \typesmi)$ (where the expectation is taken over the randomness in $\allocs(\types)$.

A mechanism is *dominant strategy incentive compatible* (DSIC) if, for all $\types \in \typespaces$, and for all $\typei' \in \typespacei$, ${\operatorname{\mathbf E}_{}{{\mathchoice{ \left [ \langle \typei, \alloc_{i}(\typei, \typesmi) \rangle  \right ]}{[ \langle \typei, \alloc_{i}(\typei, \typesmi) \rangle ]}{[ \langle \typei, \alloc_{i}(\typei, \typesmi) \rangle ]}{[ \langle \typei, \alloc_{i}(\typei, \typesmi) \rangle ]} }}} - \payi(\typei, \typesmi) \geq {\operatorname{\mathbf E}_{}{{\mathchoice{ \left [ \langle \typei,  \alloc_{i}(\typei', \typesmi) \rangle  \right ]}{[ \langle \typei,  \alloc_{i}(\typei', \typesmi) \rangle ]}{[ \langle \typei,  \alloc_{i}(\typei', \typesmi) \rangle ]}{[ \langle \typei,  \alloc_{i}(\typei', \typesmi) \rangle ]} }}} - \payi(\typei', \typesmi)$. An allocation rule $\allocs$ is said to be DSIC implementable or simply DSIC if there is a payment rule $\pays$ such that $(\allocs, \pays)$ is a DSIC mechanism. In this case, we say 