Game Theory Lecture Notes by the Gibbard Satterthwaite Theorem and Arrow's Impossibility Theorem

Note: This is a only a draft version, so there could be flaws. If you find any errors, please do send email to [email protected]. A more thorough version would be available soon in this space. 1 The Gibbard–Satterthwaite Impossibility Theorem We have seen in the last section that dominant strategy incentive compatibility is an extremely desirable property of social choice functions. However the DSIC property, being a strong one, precludes certain other desirable properties to be satisfied. In this section, we discuss the Gibbard–Satterthwaite impossibility theorem (G–S theorem, for short), which shows that the DSIC property will force an SCF to be dictatorial if the utility environment is an unrestricted one. In fact, in the process, even ex-post efficiency will have to be sacrificed. One can say that the G–S theorem has shaped the course of research in mechanism design during the 1970s and beyond, and is therefore a landmark result in mechanism design theory. The G–S theorem is credited independently to Gibbard in 1973 [1] and Satterthwaite in 1975 [2]. The G–S theorem is a brilliant reinterpretation of the famous Arrow's impossibility theorem (which we discuss in the next section). We start our discussion of the G–S theorem with a motivating example. Example 1 (Supplier Selection Problem) We have seen this example earlier (Example 2.29). We have N = {1, 2}, X = {x, y, z}, Θ 1 = {a 1 }, and Θ 2 = {a 2 , b 2 }. Consider the following utility functions (note that these are different from the ones considered in Example 2.29): u 1 (x, a 1) = 100; u 1 (y, a 1) = 50; u 1 (z, a 1) = 0 u 2 (x, a 2) = 0; u 2 (y, a 2) = 50; u 2 (z, a 2) = 100 u 2 (x, b 2) = 30; u 2 (y, b 2) = 60; u 2 (z, b 2) = 20. We observe for this example that the DSIC and BIC notions are identical since the type of player 1 is common knowledge and hence player 1 always reports the true type (since the type set is a singleton).