Abstract
In this chapter, we prove the sufficient conditions for the existence of an SRPE and an SSPE in the model described in Chap. 2. We let Assumptions 2.1–2.4 hold.
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Notes
- 1.
Note that the definition of function τ k , k ∈ E ∗ ensures that D(h) satisfies (3.8).
- 2.
The firms’ stage game payoffs are bounded, and the future stage game payoffs are discounted.
- 3.
For simplicity, the strategy profile described in the proof of Proposition 3.1 prescribes for each punished producer trading at zero prices. When the assumptions of the latter proposition are satisfied, we can construct an SRPE in which all trading takes place at positive prices. For example, a punished producer j with v j ∗ > 0 can sell his/her output at price c j (y j (a ∗ (r))) ∕ y j (a ∗ (r)) when the vector of traded quantities x(a ∗ (r)), r ∈ { 1, …, n}, is prescribed during his/her punishment. Nevertheless, this could lead to an increase in the duration of punishment, as well as to an increase in the lower bound on the values of the discount factor for which a strategy profile is an SRPE.
- 4.
The fact that we do not include the latter property among the assumptions of Lemma 1 is the only reason why we do not formulate the last sentence of the latter lemma as “Then, ξ∗ is an SSE of G.”
- 5.
The arguments in the proof also apply to a profile of mixed strategies after the following modifications. First, we replace a firm’s payoff from a profile of pure strategies occurring with a positive probability in the equilibrium by its expected payoff from its pure strategy that belongs to the support of its equilibrium mixed strategy. Second, we replace the sum of the payoffs of the members of a non-singleton coalition containing the monopsonist from a profile of pure strategies occurring with a positive probability in equilibrium by the sum of their expected payoffs from a profile of their pure strategies belonging to the Cartesian product of the supports of their mixed strategies. These remarks hold under the assumption that members of a deviating coalition can communicate the signals of their private randomizing devices to other members.
- 6.
As in the case of the maximum in (2.3), the maximum in (3.35) is well defined despite the discontinuity of the objective function caused by the disregarding of the fixed costs of the buyers in (C ∩ I)∖M with zero vector of purchases and the fixed costs of the producers in (C ∩ J)∖M with zero output. We compute the maximum for each possible subset of C∖M (including the empty set), assuming that all firms in it withdraw from the analyzed market, and then take the maximum of all such maxima.
- 7.
The argument for this claim is analogous to the one for unilateral deviations. Of course, in general the single period deviation principle does not apply to the deviations by coalitions. Nevertheless, it applies to the sum of the payoffs of the members of a deviating coalition because the latter sum is a scalar.
- 8.
When a deviation by a coalition C lasts for several periods or is infinite, in any single period, in which it takes place, it can happen that only the firms in a subcoalition Z of C intend to behave in a way that differs from the prescriptions of their equilibrium strategies. Nevertheless, in order to show that a deviation by C cannot increase the sum of the expected average discounted profits of the members of C (in a subgame in which it takes place), we have to consider all single period deviations by C (in the first period of the analyzed subgame), including those in which only the firms in the subcoalition Z of C intend to behave in a way that differs from the prescriptions of their equilibrium strategies.
- 9.
The punishments of the firms in C ∖ Z + by the firms outside C and the punishments of the firms outside C by the firms in C ∖ Z + triggered by the deviations contained in h have the same effect on the sum of the expected average discounted profits of the members of C both with and without a deviation.
- 10.
In deviations considered in (3.35), only a pair of a producer and a buyer, who belong to M, can trade a positive quantity different from the one prescribed by s ∗. If either the producer or the buyer does not belong to M, a deviation can only replace a traded quantity prescribed by s ∗ with zero.
- 11.
Example 4.3 in the following chapter also contains a stage game satisfying the assumptions of Proposition 3.4.
- 12.
Taking into account the form of production function f 4, the maximization of the sum of the profits of the members of C requires x 14 = x 24.
- 13.
The best result for a deviating coalition containing one buyer and producer 3 is worse than that for coalition {1, 4}. Producer 3 cannot trade with the buyer outside the deviating coalition. Further, his/her fixed cost exceeds the fixed cost of producer 1 by 29 > g 1(a ∗) − g 3(a ∗).
- 14.
The best result for C is better than that for coalition {3, 4, 5}. The fixed cost of producer 3 exceeds the fixed cost of producer 1 by 29 > g 1(a ∗) − g 3(a ∗). Taking into account the form of production functions f 4 and f 5, the best result for coalition {1, 3, 4, 5} is not better than the best result for C.
- 15.
By “the corresponding result” we mean the result obtained from the same trades between the members of {2, 4, 5} and firms outside C.
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© 2011 Springer-Verlag Berlin Heidelberg
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Horniaček, M. (2011). Existence of an SRPE and an SSPE. In: Cooperation and Efficiency in Markets. Lecture Notes in Economics and Mathematical Systems, vol 649. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19763-5_3
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DOI: https://doi.org/10.1007/978-3-642-19763-5_3
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