Monotonicity and Incentive Compatibility

Understanding multi-dimensional mechanism design. 1 The Model Let M = {1, . . . , m} be a finite set of agents. Every agent has private information, which can be multi-dimensional. This is called his type. The space from which an agent draws his type is called his type space. Let Ti denote the type space of agent i ∈ M . We assume Ti ⊆ R n for some integer n ≥ 1. Let T m = ×mi=1Ti. Also, denote T m −i = ×j 6=i:j∈MTj. Let t−i = (t1, . . . , ti−1, ti+1, . . . , tm}. The set of outcomes or alternatives is denoted by A. The valuation of agent i is a mapping vi : A× Ti → R. An allocation rule is a mapping f : T m → A. A payment rule is a mapping p : T m → R. An allocation rule along with a payment rule is called a mechanism. A classical example of multi-dimensional mechanism design problem is the design of auctions for selling multiple objects. If there are k objects, every agent (buyer) has a 2 dimensional type space, denoting his value for every possible bundle of objects. An outcome says which objects are assigned to which agents. Definition 1 An allocation rule f is dominant strategy incentive compatible (DSIC) if there exists a payment rule p such that for every agent i ∈ M and for every t−i ∈ T m −i, we have vi(f(ti, t−i), ti) − p(ti, t−i) ≥ vi(f(si, t−i), ti) − p(si, t−i) ∀ si, ti ∈ Ti. (1) Sincere thanks to Rudolf Müller for discussions that led to the development of the first version of these notes in Maastricht University. Subsequent versions continue to benefit from discussions with Mridu Prabal Goswami and Arunava Sen. Presenting the contents of the notes to the reading group at ISI Delhi is improving these notes every day. Indian Statistical Institute, 7 Shaheed Jit Singh Marg, New Delhi 110016, India