Rent Extraction through Alternative Forms of Competition in the Provision of Paternalistic Goods

Table 1

Revelation for player 𝐴 in the independent types case.

	0≤𝐸(𝛼𝐵)≤𝑠	𝑠≤𝐸(𝛼𝐵)≤3𝑠	𝐸(𝛼𝐵)≥3𝑠
𝛼𝐴	[0,𝑠]	30,2𝐸𝛼𝑠−^𝐵2	[0,𝐸(𝛼𝐵))−3𝑠]
𝛼^∗𝐴	0	0	𝛼𝐴
𝛼𝐴	[𝑠,3𝑠−2𝐸(𝛼𝐵)]	32𝐸𝛼𝑠−^𝐵2,𝐸𝛼𝐵+3𝑠	[𝐸(𝛼𝐵)−3𝑠,𝐸(𝛼𝐵)+3𝑠]
𝛼^∗𝐴	12(𝛼𝐴−𝑠)	13(2𝛼𝐴+𝐸(𝛼𝐵))−𝑠	13(2𝛼𝐴+𝐸(𝛼𝐵))−𝑠
𝛼𝐴	[3𝑠−2𝐸(𝛼𝐵),𝐸(𝛼𝐵)+3𝑠]	[𝐸(𝛼𝐵)+3𝑠,+∞)	[𝐸(𝛼𝐵)+3𝑠,+∞)
𝛼^∗𝐴	13(2𝛼𝐴+𝐸(𝛼𝐵))−𝑠	12(𝛼𝐴+𝐸(𝛼𝐵)−𝑠)	12(𝛼𝐴+𝐸(𝛼𝐵)−𝑠)
𝛼𝐴	[𝐸(𝛼𝐵)+3𝑠,+∞)
𝛼^∗𝐴	12(𝛼𝐴+𝐸(𝛼𝐵)−𝑠)