Cheap Talk, Monitoring and Collusion

Many collusive agreements involve the exchange of self-reported sales data between competitors, which use them to monitor compliance with a target market share allocation. Such communication may facilitate collusion even if it is unverifiable cheap talk and the underlying information becomes publicly available with a delay. The exchange of sales information may allow firms to implement incentive-compatible market share reallocation mechanisms after unexpected swings, limiting the recourse to price wars. Such communication may allow firms to earn profits that could not be earned in any collusive, symmetric pure-strategy equilibrium without communication.

1. Harrington (2006), Levenstein and Suslow (2006). See in particular the developments in these papers on the lysine, copper plumbing, zinc phosphate, citric acid and vitamins cartels.

2. Reliable information can come from companies’ annual reports of from import statistics, or it can be collected and disseminated by a professional association—in some cases with the help of independent auditors. In several recent cases, market share information was found to become available with a delay of about one year.

3. Athey and Bagwell (2001, 2008); see also Aoyagi (2007), Harrington (2017).

4. Compte (1998), Kandori and Matsushima (1998), Obara (2009). See also Rahman (2014).

5. Athey and Bagwell (2001, 2008) also derive comparative statics results on the role of communication. But communication in their models is pre-play coordination meant to account for private information on costs.

6. On the general analysis of repeated games in which information on other players’ actions is revealed with lags, see (Abreu et al. 1991), and (Fudenberg et al. 2014) on the impact of cheap-talk communication in such games. Igami and Sugaya (2021) estimate the impact of the lag on the possibility of collusion on the basis of a stylized model of the vitamins cartel.

7. Another related paper is Mouraviev (2014), which shows that communication can increase collusive profits, albeit in an environment that is very different from ours since firms can communicate verifiable information and the only limit on communication comes from the risk of being caught.

8. Several recent papers—such as Acemoglu et al. (2009) and Gentzkow and Kamenica (2017)—focus on equilibria that involve pure strategies along all equilibrium paths—but allow however mixed strategies off-equilibrium. It must be acknowledged however that collusive behavior that is based on mixed strategies along the equilibrium path seems to be observed in some industries (Wang 2009)—in line with theoretical results that show that mixing can improve efficiency (Kandori and Obara 1998).

9. Models of collusion often make this assumption (see the discussion in Athey et al. (2004)).

10. The above reasoning works unless total demand in period 1 is such that a firm would be able to infer that it had a 100% market share from observing its own sales, without any need for other firms’ sales data. This is the case with probability less than or equal to A, hence the factor \((1-A)\) in (1).

11. As is explained below, transition rules are a bit more complex than this summary description, to account for the possibility that during a correction phase at the expense of some firm, another firm may benefit from biased demand and become the target of a new correction phase.

12. This remark is at odds with the oft-made claim that “aggregating the data [on firms’ historical and current prices, costs, and output] largely removes the value of information in facilitating collusion” (Carlton et al. 1997). Awaya and Krishna (2020) present another mechanism through which the communication of credible aggregate sales data to individual firms can facilitate collusion: In their model (unlike in their 2016 and 2019 papers) firms have an incentive to misreport sales, hence the need for ’Swiss accountants’. It must be noted that in some models, more accurate information may decrease the set of attainable profits because it may help firms to identify more profitable deviations (Sugaya and Wolitzky 2018a). However, such a mechanism cannot be present in our model, since the deviations considered here do not rely on the deviating firm’s information on other firms’ sales.

13. See Kühn (2001).

14. Official Journal of the European Union, C 11/91, 14.1.2011, Communication from the Commission — Guidelines on the applicability of Article 101 of the Treaty on the Functioning of the European Union to horizontal co-operation agreements.

15. Kühn (2011) advocates “an analysis of the marginal impact of the information exchange on monitoring or the scope for coordination in the market. If the marginal impact appears small, the case should be closed.”

16. See in particular Pai, Roth and Ullman (2016) and Sugaya and Wolitzky (2017, 2018b). Awaya and Krishna (2016, 2019) also overcome this difficulty by characterizing an upper bound on collusive profits in the no-communication case under quite general assumptions.

Authors and Affiliations

1. Paris School of Economics and CNRS, 48, Boulevard Jourdan, 75014, Paris, France

   David Spector

Corresponding author

Correspondence to David Spector.

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Appendix

Proof of Proposition 1 We introduce the following notations: \(\lambda _{1}\) denotes \(\underset{}{Max}\left( \underset{Q\in \left( 0,S_{max}\right) }{Max}\frac{\mu ^{N}(Q)}{\mu ^{B}(Q)},\underset{Q\in \left( 0,S_{max}\right) }{Max}\frac{\mu ^{B}(Q)}{\mu ^{N}(Q)}\right)\) and \(\lambda _{2}\) denotes \(\underset{Q\in \left( 0,\frac{S_{max}}{n}\right) }{Max}\frac{\mu ^{N}(Q)}{\mu ^{N}(nQ)}\).

We consider a hypothetical PSSE \(Eq*\) such that the expected sum of all firms’ future discounted profits is greater than \(\frac{(V-c)D}{1-\delta }(1-\sigma )\) for some small \(\sigma\). The core of this proof is the calculation of a lower bound on the profits that some firms—say Firm 1—could earn by following a certain strategy, which can be interpreted as undercutting in each of the first four periods. This requires intermediate results about the strategies that are followed by Firms 2 to n- conditional on the information available to them.

Step 1. Minimum combined profits after certain histories. We define the history of the game until period t as the prior sequence of prices and sales (including period t). We provide hereafter a lower bound on the expectation of total period t profits conditional on past history’s belonging to some set H. Let Prob(H) denote the probability, according to \(Eq*\), that the history of the game until period \((t-1)\) belongs to H (and let it be equal to 1 if \(t=1\)). Since total expected profits in any period cannot exceed \((V-c)D(1+\gamma ^{B}u)\), the expected discounted sum of total profits in all periods cannot exceed \((V-c)D(1+\gamma ^{B}u)\left( \frac{1}{1-\delta }-\delta ^{t-1}Prob(H)\right) +\delta ^{t-1}Prob(H)\prod (H)\), where \(\prod (H)\) denotes expected total period t profits conditional on past history belonging to H. This expected discounted sum is greater than \(\frac{(V-c)D(1-\sigma )}{1-\delta }\) , which implies that \(\prod (H)>(V-c)D(1-\alpha _{t}(Prob(H),\sigma ))\) with \(\alpha _{t}(x,\sigma )=\frac{\sigma +\gamma ^{B}u}{x(1-\delta )\delta ^{t-1}}\).

Step 2. A quasi-lower bound on equilibrium prices. Assume that in period i, conditional on a certain event (a set of past histories), the expectation (according to \(Eq*\)) of total profits \(\prod\) is greater than \((V-c)D(1-\alpha )\) for some \(\alpha\). Defining \(g(\alpha )=\sqrt{\frac{\alpha +\gamma ^{B}u}{1-\gamma ^{B}}}\), the probability (conditional on that same event) that the lowest of all n prices in period i is greater than \(c+(V-c)(1-g(\alpha ))\) is greater than \(1-g(\alpha )\). Proof: Total expected profits cannot exceed \(D\left( Min\left( p_{t}^{1},...,p_{t}^{n},V\right) -c\right)\) if demand is normal and \((V-c)D(1+u)\) otherwise. If the probability that \(Min\left( p_{t}^{1},...,p_{t}^{n}\right) -c<(V-c)\left( 1-g(\alpha )\right)\) is greater than \(g(\alpha )\), then \(\frac{\prod }{(V-c)D}<\gamma ^{B}(1+u)+(1-\gamma ^{B})\left( 1-g(\alpha )+g(\alpha )(1-g(\alpha ))\right) =1-\alpha\).

Step 3. The profit from deviating in period 1. Since \(Eq*\) is symmetric, it prescribes all firms to set the same price \(p_{1}^{*}\) in period 1. The total expected profit induced by \(Eq*\) in period 1 is thus less than or equal to \(D\left( p_{1}^{*}-c\right) ,\) which implies that \(p_{1}^{*}-c>\left( V-c\right) \left( 1-g_{1}\right)\), with \(g_{1}=\alpha _{1}(1,\sigma )\). If Firm 1 deviates and sets a price \(p_{1}^{dev}=c+(V-c)\left( 1-g_{1}\right)\) in period 1, it serves the entire demand if demand is normal or biased in its favor, which happens with a probability that exceeds \(1-\gamma ^{B}\).

Step 4. Firm 1’s possible deviation in period 2. Since \(Eq*\) is symmetric, there exists \(p_{2}^{*}(0)\) such that \(Eq*\) prescribes a firm that sold zero in period 1 to set price \(p_{2}^{*}(0)\) in period 2. The equilibrium probability that all firms sell zero in period 1 is \(\gamma ^{L}\). Step 1 implies that \(p_{2}^{*}(0)>c+\left( V-c\right) \left( 1-\alpha _{2}(\gamma ^{L},\sigma )\right)\). Let \(g_{2}\) denote \(\alpha _{2}(\gamma ^{L},\sigma )\). If Firm 1 deviates in period 2 by setting a price \(p_{2}^{dev}=c+(V-c)\left( 1-g_{2}\right)\) and all other firms made zero profits in period 1, its expected period 2 profit is greater than \((V-c)D\left( 1-g_{2}\right) \left( 1-\gamma ^{B}\right)\).

Step 5. Firm 1’s possible deviation in period 3. For each Q in \(\left( 0,S_{max}\right]\), let \(p_{3}^{*}(Q)\) denote the price that is prescribed by \(Eq*\) in period 3 for a firm that has observed that (i) it sold zero in period 2; and (ii) in period 1, one of the firms (not itself) sold Q while all others sold 0. We also define \(g_{3}^{*}(Q)\) by the identity \(p_{3}^{*}(Q)-c=\left( V-c\right) (1-g{}_{3}^{*}(Q)).\) The equilibrium probability that (i) Firms 2 to n had zero sales in periods 1 and 2 and (ii) Firm 1 had nonzero sales in period 1 is greater than \(\frac{\gamma ^{B}}{n}\gamma ^{L}\). Steps 1 and 2 imply that conditional on (i) and (ii), with probability greater than \(1-g\left( \alpha _{3}(\frac{\gamma ^{B}}{n}\gamma ^{L},\sigma )\right)\), the minimum of all prices in period 3 is above \(c+(V-c)\left( 1-g\left( \alpha _{3}(\frac{\gamma ^{B}}{n}\gamma ^{L},\sigma )\right) \right)\): There exists \(S_{3}\subset \left( 0,S_{max}\right)\) such that (i) \(\mu ^{B}(S_{3})>1-g\left( \alpha _{3}(\frac{\gamma ^{B}}{n}\gamma ^{L},\sigma )\right)\) and (ii) \(\forall Q\in S{}_{3},\;g{}_{3}^{*}(Q)<g_{3}.\) Also, by the definition of \(\lambda _{1}\), \(\mu ^{N}(S_{3})>1-\lambda _{1}g\left( \alpha _{3}(\frac{\gamma ^{B}}{n}\gamma ^{L},\sigma )\right)\). Define \(p_{3}^{dev}=c+(V-c)\left( 1-g_{3}\right)\) with \(g_{3}=g\left( \alpha _{3}(\frac{\gamma ^{B}}{n}\gamma ^{L},\sigma )\right)\). Conditional on Firms 2 to n having sold zero in periods 1 and 2 and on Firm 1’s having made nonzero sales in period 1 (which happens with probability greater than \(\left( 1-\gamma ^{B}\right) \left( 1-\gamma ^{B}-\gamma ^{L}\right)\) if Firm 1 deviated in the first two periods), if Firm 1 sets price \(p_{3}^{dev}\) in period 3, this price is lower than all other firms’ prices with probability greater than \(1-\lambda _{1}g_{3}\), which yields an expected profit greater than \((V-c)\left( 1-g_{3}\right) \left( 1-\lambda _{1}g_{3}\right) \left( 1-\gamma ^{B}\right)\).

Step 6. The period 2 prices set by a firm observing its period 1 sales belonged to\(\left( 0,\frac{S_{max}}{n}\right)\) , according to \(Eq*\) .

For each \(Q\in \left( 0,\frac{S_{max}}{n}\right)\) let \(p_{2}^{*}(Q)\) denote the equilibrium price set in period 2 by a firm that observes that it sold Q in period 1. We define the set \(S'\) as follows: \(S'=\left\{ Q\;s.t.\;Q\in \left( 0,\frac{S_{max}}{n}\right) \;and\;p_{2}^{*}(Q)>V\right\} .\) With probability \(\left( 1-\gamma ^{B}\right) \mu ^{N}(nS')\) (with \(nS'\) denoting the set of all elements of \(S'\) multiplied by n), demand is normal and each firm sells some \(Q\in S'\) in period 1, which implies that all firms set a price that is strictly above V in period 2, which leads to expected period 2 profits that are smaller than or equal to \(\gamma ^{B}(V-c)D(1+u)\). Step 1 implies therefore that \(1-\alpha _{2}\left( \left( 1-\gamma ^{B}\right) \mu ^{N}(nS'),\sigma \right) <\gamma ^{B}(V-c)D(1+u)\), or, after rearranging terms, \(\mu ^{N}(nS')<\frac{\sigma +\gamma ^{B}u}{\left( 1-\gamma ^{B}\right) \left( 1-\gamma ^{B}(1+u)\right) \delta }\), which implies \(\mu ^{N}(S')<\lambda _{2}\frac{\sigma +\gamma ^{B}u}{\left( 1-\gamma ^{B}\right) \left( 1-\gamma ^{B}(1+u)\right) \delta }\).

Step 7. The probability, according to \(Eq*\) , that Firm 1’s sales belong to \(\left( 0,\frac{S_{max}}{n}\right) \setminus S'\) in period 1 and to \(\left( 0,S_{max}\right)\) in period 2 while all other firms sell zero in periods 1 and 2. According to \(Eq*\), the probability that Firm 1’s period 1 sales belong to \(\left( 0,\frac{S_{max}}{n}\right) \setminus S'\) whereas all other firms’ period 1 sales are zero is \(\frac{\gamma ^{B}}{n}\left( 1-\mu ^{B}(S')-\mu ^{B}\left( \frac{S_{max}}{n},S_{max}\right) \right)\). If Firm 1’s period 1 sales belong to \(\left( 0,\frac{S_{max}}{n}\right) \setminus S'\) whereas all other firms’ period 1 sales are zero, then according to \(Eq*\), in period 2 Firm 1 sets a price \(p_{2}^{*}(Q)\le V\) and all other firms set a price \(p_{2}^{*}(0)>\left( V-c\right) \left( 1-g_{2}\right) +c\) (by Step 4), so that \(p_{2}^{*}(Q)-p_{2}^{*}(0)<(V-c)g_{2}\), which implies that the entirety of period 2 demand goes to Firm 1 if demand is biased in Firm 1’s favor and consumers’ valuation v of Firm 1’s product is such that \(\frac{v-c}{V-c}-1>g_{2}\), which is the case with probability \(\frac{\gamma ^{B}}{n}\nu \left( \left( (V-c)(1+g_{2})+c,V'\right) \right) =\frac{\gamma ^{B}}{n}\left( 1-\frac{g_{2}}{u}\right)\) if \(g_{2}<u\), or \(\frac{\sigma +\gamma ^{B}u}{\gamma ^{L}(1-\delta )\delta }<u\). From here onwards, we assume that \(\frac{\gamma ^{B}}{\gamma ^{L}}\) is small, and we consider values of \(\sigma\) that are small relative to u, so that \(g_{2}<u\). The probability according to \(Eq*\) that firms other than Firm 1 sell zero in the first and the second period whereas Firm 1’s sales in these periods belong respectively to \(\left( 0,\frac{S_{max}}{n}\right) \setminus S'\) and \(\left( 0,S_{max}\right)\) is thus greater than \(\left( \frac{\gamma ^{B}}{n}\right) ^{2}\left( 1-\mu ^{B}(S')-\mu ^{B}\left( \frac{S_{max}}{n},S_{max}\right) \right) \left( 1-\frac{g_{2}}{u}\right)\). \(\mu ^{B}(S')<\lambda _{1}\mu ^{N}(S')\) so that (using the results of Step 6) the latter expression is greater than \(\left( \frac{\gamma ^{B}}{n}\right) ^{2}\left( 1-\lambda _{1}\lambda _{2}\frac{\sigma +\gamma ^{B}u}{\left( 1-\gamma ^{B}\right) \left( 1-\gamma ^{B}(1+u)\right) \delta }-A\right) \left( 1-\frac{g_{2}}{u}\right)\).

Step 8. The period 4 prices set by all other firms after having sold zero in periods 1 to 3 and observing that Firm 1’s sales belong to \(\left( 0,\frac{S_{max}}{n}\right) \setminus S'\) in period 1 and \(\left( 0,S_{max}\right)\) in period 2. The probability, according to \(Eq*\), that Firms 2 to n sell zero in periods 1, 2, and 3, while Firm 1 makes nonzero sales in periods 1 and 2, and its period 1 sales belong to \(\left( 0,\frac{S_{max}}{n}\right) \setminus S'\), is greater than the latter lower bound times \(\gamma ^{L}\). Define

Step 2 implies that conditional on Firm 1’s having nonzero sales in periods 1 and 2 (belonging to \(\left( 0,\frac{S_{max}}{n}\right) \setminus S'\) in period 1) and other firms having zero sales in periods 1 to 3, then with probability greater than \(1-g_{4},\) Firms 2 to n set a price that is greater than \(p_{4}^{dev}=c+(V-c)\left( 1-g_{4}\right)\). Notice that the probability that after Firm 1 deviated by setting prices \(p_{i}^{dev}\) in period i (\(i=1,2,3)\), its sales belonged to \(\left( 0,\frac{S_{max}}{n}\right) \setminus S'\) in period 1 and \(\left( 0,S_{max}\right)\) in period 2 while other firms sold zero in periods 1 to 3 is greater than \(\left( 1-\mu ^{N}(S')-\mu ^{N}\left( \frac{S_{max}}{n},S_{max}\right) -\frac{\gamma {}^{L}}{1-\gamma {}^{B}}\right)\) \(\left( 1-\gamma ^{B}\right) ^{3}\), which is greater than \(\left( 1-\lambda _{2}\frac{\sigma +\gamma ^{B}u}{\left( 1-\gamma ^{B}\right) \left( 1-\gamma ^{B}(1+u)\right) \delta }-A-\frac{\gamma {}^{L}}{1-\gamma {}^{B}}\right) \left( 1-\gamma ^{B}\right) ^{3}\).

Step 9. Firm 1’s possible deviations in periods 1 to 4 and the corresponding expected profits. Steps 1 to 8 imply that if Firm 1 sets price \(p_{i}^{dev}\) in period i (\(i=1,2,3,4)\) then its expected profit is at least \((V-c)D\left( 1-g_{1}\right) \left( 1-\gamma ^{B}\right)\) in period 1, \(\left( V-c\right) D\left( 1-g_{2}\right) \left( 1-\gamma ^{B}\right) ^{2}\) in period 2, \(\left( V-c\right) D\left( 1-\lambda _{1}g_{3}\right) \left( 1-g_{3}\right) \left( 1-\gamma ^{B}-\gamma ^{L}\right) \left( 1-\gamma ^{B}\right) ^{2}\) in period 3, and \(\left( V-c\right) D\left( 1-\lambda _{2}\frac{\sigma +\gamma ^{B}u}{\left( 1-\gamma ^{B}\right) \left( 1-\gamma ^{B}(1+u)\right) \delta }-A-\frac{\gamma {}^{L}}{1-\gamma {}^{B}}\right) \left( 1-\gamma ^{B}\right) ^{4}\left( 1-g_{4}\right) ^{2}\) in period 4.

Step 10. (1) implies the existence of a profitable deviation. Assume that inequality (1) holds: \(1+\delta +\delta ^{2}+(1-A)\delta ^{3}>\frac{1}{n\left( 1-\delta \right) }\) . (1) is equivalent to \(\delta ^{4}+A\delta ^{3}<1-\frac{1}{n}\), which provides a strictly positive lower bound on the possible values of \(\frac{1}{1-\delta }\). If \(\gamma ^{L}\), \(\gamma ^{B}\), \(\frac{\gamma ^{B}u}{\gamma ^{L}}\), \(\frac{nu}{(1-\delta )\delta ^{3}\gamma ^{B}\gamma ^{L}(1-A)}\) and \(\sigma\) are close to zero, then: \(g_{1}\), \(g_{2}\), \(g_{3}\), and \(g_{4}\) are close to zero; each \(p_{i}^{dev}\) (\(i=1,2,3,4)\)is close to V; and with a probability close to 1 in periods 1 to 3, and close to \(1-A\) in period 4, a firm that sets price \(p_{i}^{dev}\) in period i (\(i=1,2,3,4)\) serves the entire demand in each of the first four periods. Such a deviation yields an expected profit that is close to \((V-c)D\left( 1+\delta +\delta ^{2}+(1-A)\delta ^{3}\right)\). If \(1+\delta +\delta ^{2}+(1-A)\delta ^{3}>\frac{1}{n\left( 1-\delta \right) }\) and \(\gamma ^{B}u\) is close to zero (which follows from \(\frac{\gamma ^{B}}{\gamma ^{L}}\)and \(\frac{nu}{(1-\delta )\delta ^{3}\gamma ^{B}\gamma ^{L}(1-A)}\) being close to zero), then this is greater than a firm’s expected profit according to \(Eq*\), which cannot exceed \(\frac{(V-c)D(1+\gamma ^{B}u)}{n\left( 1-\delta \right) }\).

Proof of Proposition 2

Preamble: the parameters of the price war. We assume that condition (2) holds (which implies \(\delta >\frac{1}{2}\)), and that \(\gamma ^{B}\), \(\gamma ^{L}\) and u are close to zero, and \(c>5(V'-V)\). We want to find price-war parameters such that: (i) any firm’s expectation of the sum of its future discounted profits at the start of a price war is 0 if all firms are expected to follow the strategies that are described in Section 5.1:\((V'+1)\); and (ii) these strategies cannot be individually improved upon. Let \(PW_{r}(p,k')\) denote a firm’s expectation of the discounted sum of its future payoffs at the start of the r-th period of a price war lasting \(k'\) periods (\(1\le r\le k'\)) with price p (\(p<V\) ), on the assumption that all firms act according to the candidate equilibrium. At the beginning of the r-th period of a price war lasting \(k'\) periods, each firm’s current period expected profit is \(\frac{\left( p-c\right) D}{n}\), and with probability \(\left( \gamma ^{B}+\gamma ^{L}\right)\), the state of the world in the next period is the first period of a price war; whereas with probability \(1-\left( \gamma ^{B}+\gamma ^{L}\right)\), it is ’\((r+1)\)-th period of a price war’ if \(r<k'\) and ’normal collusion’ if \(r=k'\). This implies the following equalities:

which implies

Let \(F(p,k')\) denote the right-hand side of this equation, multiplied by n/D:

If \(\delta >\frac{1}{2}\) and \(\gamma ^{B}\), \(\gamma ^{L}\), and u are close enough to zero, then \(F\left( c-(V'-V),1\right) >0\). Also, \(\underset{\kappa \rightarrow \propto }{Lim}F\left( c-(V'-V),\kappa \right) <0\). Therefore, if \(\delta >\frac{1}{2}\) and \(\gamma ^{B}\), \(\gamma ^{L}\), and u are close enough to zero, there exists an integer \(k'\ge 1\) such that \(F\left( c-(V'-V),k'+1\right)<0<F\left( c-(V'-V),k'\right)\). We show now that \(F\left( c-5(V'-V),k'\right) <0\):

But

which is greater than 1 if \(\delta >\frac{1}{2}\) and \(\gamma ^{B}\) and \(\gamma ^{L}\) are both close to zero, which imples that \(F\left( c-5(V'-V),k'\right) <0\). By continuity, there exists some price \(p^{w}\) between \(c-5(V'-V)\) and \(c-(V'-V)\) such that \(PW_{1}(p^{w},k')=F(p^{w},k')=0\).

We now prove that at the beginning of any stage of a price war, it is a best response for a firm to set price \(p^{w}\) and truthfully report its sales, given that all firms are expected to follow the strategies that are described above. First, the Bellman equation above implies that for all r such that \(1<r\le k'\), \(PW_{r}(p^{w},k')>PW_{1}(p^{w},k')=0\).

Consider a firm at the start of the r-th period of a price war. Complying with the candidate equilibrium strategy yields an expectation of the sum of future discounted profits that is equal to \(PW_{r}(p^{w},k')\ge 0\). Let \(BR_{r}(p^{w},k')\ge 0\) denote the expectation of the sum of future discounted profits of a firm (say, Firm 1) at the beginning of the r-th period of a price war, on the assumption that the firm, from period 1 onwards, maximizes the expectation of the discounted sum of its future profits, and that all other firms act according to the candidate equilibrium.

First, at the beginning of any price war period, a firm cannot earn a strictly positive profit: This would require a price above c and therefore greater than \(p^{w}+(V'-V)\), implying zero sales.

Second, if Firm 1 sets a price that is different from \(p^{w}\), then with probability 1, the state of the world in the next period is ’first period of a price war’. This is because when not all firms set the same price, it is impossible that all firms have identical nonzero sales and whatever Firm 1’s announcement, the probability that all firms report identical nonzero sales is zero.

This implies \(BR_{1}(p^{w},k')\le \delta BR_{1}(p^{w},k')\), which implies that \(BR_{1}(p^{w},k')\le 0\) and therefore \(BR_{1}(p^{w},k')=0\). It is therefore a best response in period 1 to set a price that is equal to \(p^{w}.\) If the best response in the r-th period of a price war (\(r>1\)) involved a price different from \(p^{w}\), then the above analysis implies \(BR_{r}(p^{w},k')\le \delta BR_{1}(p^{w},k')=0<BR_{r}(p^{w},k'),\) which is a contradiction. Therefore, the price war that is described above is indeed an equilibrium. The price war that is mentioned in the remainder of this proof is the one that is described above: with \(PW_{1}(p^{w},k')=0\).

Step 1: the expected sum of future discounted profits for a firm according to the state of the game. Let \(W_{c,r}\), \(W_{c-,r}\) and W denote the expected sum of a firm’s future discounted profits at the beginning of, respectively, a correction period at its own expense with r remaining periods (\(1\le r\le k\)), a correction period at another firm’s expense with r remaining periods, and a normal collusion period (on the assumption that all firms behave according to the candidate equilibrium). The assumptions of the model imply \(W=\frac{(V-c)D}{n(1-\delta )}\). Also, since ahead of any period along any equilibrium path, n or \(\left( n-1\right)\) firms are in a symmetric situation, \(W_{c-,r}\le \frac{(V-c)D}{(n-1)(1-\delta )}\).

In any correction period, there is a probability \(\frac{(n-1)\gamma ^{B}}{n}\) that one of the non-targeted firms will benefit from biased demand. Therefore, for any r between 1 and k, \(W_{c,r}=\delta \left[ \left( 1-\frac{(n-1)\gamma ^{B}}{n}\right) W_{c,r-1}+\frac{(n-1)\gamma ^{B}}{n}W_{c-,k}\right]\) (with the notation \(W_{c,0}=W\)) and

The above results imply that for any number of firms \(n\ge 3\), the following inequalities hold:

We prove now that for any r (\(1\le r\le k\)), \(W_{c,r}\le W\le W_{c-,r}\). First, we show by induction that either for all r, \(W_{c,r}\le W\), or for all r, \(W_{c,r}\ge W\). Assume (by contradiction) that \(W_{c,k}>W\). The equality \(W_{c,k}+(n-1)W_{c-,k}=nW\) implies \(W_{c-,k}<W\). This and the equation that characterizes \(W_{c,1}\) implies \(W_{c,1}<W\), which is a contradiction. Finally, the inequality \(W_{c,r}\le W\) implies that for any r (\(2\le r\le k\)), \(W_{c,r+1}\le W_{c,r}\) and \(W_{c-,r+1}\ge W_{c-,r}\).

Step 2. Whatever the state of the world at the beginning of period \(t\), if Firm i complied with the strategies that are prescribed by the candidate equilibrium in all previous periods and it assumes that all other firms behave according to the candidate equilibrium, and Firm i did set a price at the beginning of period t in accordance with the candidate equilibrium, or the observation of its own sales combined with the knowledge of the price it set at the beginning of the period allows it to know that the distribution of current period sales will not reveal any deviation, then reporting sales truthfully is a best response for Firm i.

Proof. Since misreporting at the end of period t is detected at the latest at the end of period (\(t+1\)), which leads to a price war from period (\(t+2\)) onwards, misreporting would allow Firm i to earn at most \((V-c)D(1+\gamma ^{B}u)\) in period \((t+1)\) and an expected discounted sum of subsequent profits equal to 0. Complying with the strategy prescribed by the candidate equilibrium would lead, at the beginning of period \((t+1)\), to an expected sum that is greater than or equal to \(W_{c,k}\). Since (4) implies \(W_{c,r}\ge (V-c)D(1+\gamma ^{B}u)\) for all \(r\) if \(\gamma ^{B}\) and u are close enough to zero (in which case it boils down to \(\delta ^{k}\ge n(1-\delta )\)), truthful sales reporting is a best response.

Step 3. Notation: \(D_{t}\) denotes total demand in period t. At the beginning of a correction period, it is a best response for all firms to set the price that is prescribed by the candidate equilibrium.

Proof

Assume that the state of the game at the beginning of period t is ’correction at the expense of Firm 1 with r remaining periods’. We prove first that it is optimal for Firm 1 to behave as prescribed by the candidate equilibrium. We showed (Step 2) that conditional on setting a price that is equal to \(V'+1\) it is optimal for Firm 1 to report its zero sales truthfully. Assume that Firm 1 sets \(p_{t}^{1}\ne V'+1\). Any price greater than \(V'\) yields the same payoff distribution for Firm 1 in period t and subsequent periods as \(p_{t}^{1}=V'+1\). Consider now a price \(p\in (V,V')\). Such a price leads to exactly the same outcome as \(p_{t}^{1}=V'+1\) unless demand is biased in favor of Firm 1, with a valuation \(v\ge p\). In this latter case, if Firm 1 sets \(p_{t}^{1}=V'+1\), it earns zero in period t, and its expected sum of future profits is at least \(\delta W_{c,r-1}\) ; whereas if it sets \(p_{t}^{1}=p\), its deviation will be detected at the end of the following period, which leads to profits that are below \((V-c)D\left[ (1+ \delta )(1+\gamma ^{B}u) \right]\). (2) implies \(\delta W_{c,r-1}>(V-c)D\left[ (1+ \delta (1+\gamma ^{B}u) \right]\) if \(\gamma ^{B}\) and u are close enough to zero, which makes such a deviation unprofitable.

Consider now a price \(p\le V\). Unless demand is zero or biased, setting such a price leads Firm 1 to being ’exposed’ after two periods at most. The corresponding expected sum of future discounted profits is thus less than \((V-c)D(1+\gamma ^{B}u)\left[ (1+ \delta ) +\left( \gamma ^{L}+\gamma ^{B}\right) \right]\). If \(\gamma ^{B}\), \(\gamma ^{L}\) and u are close enough to zero, (2) implies that this expression is less than \(W_{c,k}\): less than \(W_{c,r}\) for any \(r\le k\). Therefore, it is optimal for Firm 1 to follow the strategy that is prescribed by the candidate equilibrium.

Consider now a firm other than Firm 1—say, Firm 2. Complying with the actions that are prescribed by the candidate equilibrium leads for Firm 2 to an expected sum of future discounted profits that is equal to \(W_{c-,r}\). Assume now that Firm 2 deviates and sets a price \(p_{t}^{2}\ne V\). If \(p_{t}^{2}>V\), then Firm 2 earns zero in period t and whatever it reports at the end of period t, its deviation is detected at the end of period \((t+1)\) unless demand in period t is zero, because a distribution of sales such that only two firms have zero sales is incompatible with equilibrium. Therefore the corresponding expected sum of future discounted profits is less than or equal to \((V-c)D(1+\gamma ^{B}u)\left[ \delta +\frac{\gamma ^{L}\delta ^{2}}{1-\delta }\right] .\) If \(\gamma ^{B}\), \(\gamma ^{L}\), and u are close enough to zero, (3) implies that this expression is less than \(W_{c,k}\), and therefore less than \(W_{c-,r}\). A price \(p_{t}^{2}>V\) therefore cannot improve upon the behavior that is prescribed by the candidate equilibrium for Firm 2.

Consider now the possibility of a price \(p_{t}^{2}<V\). Such a price would yield Firm 2 at most \((V-c)D\) in expectation in period t. If demand in period t is zero, which happens with probability \(\gamma ^{L}\), period t sales are identical to what they would be absent a deviation by Firm 2. Therefore it would be optimal for Firm 2 in this case to report zero sales (Step 2), which leads in period \((t+1)\) to either ’normal collusion’ (if \(r=1)\) or to ’correction at the expense of Firm 1, with (\(r-1\)) remaining periods’ (if \(r\ge 2\)). This would lead at the beginning of period \((t+1)\) to an expected sum of future discounted profits that is equal to \(W_{c-,r-1}\) (with the notation \(W_{c-,{\text {0}}}=W\)). If demand is normal, a deviation leads to a sales profile that is compatible with equilibrium behavior (with demand biased in favor of Firm 2). It is then optimal for Firm 2 to reveal its sales truthfully (Step 2), which leads at the beginning of period \((t+1)\) to an expected sum of future discounted profits that is equal to \(W_{c,k}\). Finally, if demand is biased, then with probability 1 one of the firms serves the entire demand, which leads to an expected sum of future discounted profits that is below \(W_{c-,k}\). Therefore, a deviation with \(p_{t}^{2}<V\) would lead at the beginning of period t to an expected sum of Firm 2’s future discounted profits that is less than or equal to \((V-c)D+\delta \left( \gamma ^{L}W_{c-,r-1}+\left( \left( 1-\gamma ^{B}\right) -\gamma ^{L}\right) W_{c,k}+\gamma ^{B}W_{c-,k}\right)\), whereas in the absence of deviation this expected sum is equal to \(W_{c-,r}\). The difference between this expected sum in the absence and in the presence of such a deviation is thus greater than or equal to

If \(\gamma ^{B}\) is close enough to zero, (2) implies that this difference is positive, so that it is a best response for Firm 2 to follow the prescribed equilibrium strategy.

Step 4. If the state of the world at the beginning of period \(t\) is ’normal collusion’ and all firms followed the strategy prescribed by the candidate equilibrium in previous periods, then setting \(p_{t}^{1}=V\) is a best response for Firm 1.

Proof

Consider a subgame that starts in period \(t\), s. t. the state of the game at the beginning of period \(t\) is ’normal collusion’. Firm 1’s expected sum of future discounted profits is W if it sets \(p_{t}^{1}=V\). We prove hereafter by contradiction that if (2)-(3) hold, then in a ’normal collusion’ period, a price such that \(p_{t}^{1}<V\) or \(p_{t}^{1}>V\) leads to an expected sum of future discounted profits that is smaller than or equal to W.

We assume the existence of a best response such that with a positive probability Firm 1 sets a price \(p_{t}^{1}<V\) in some ’normal collusion’ state. Let \(W'\) denote Firm 1’s expected sum of future discounted profits at the beginning of any ’normal collusion’ state, given this (or any other) best response.

In period \(t\), setting a price \(p_{t}^{1}<V\) allows Firm 1 to serve the entire demand unless demand is biased in favor of some other firm. If \(D_{t}=0\), then the state of the world is ’normal collusion’ again in period \((t+1)\), which leads to an expected sum of future discounted profits that is equal to \(W'\) at the beginning of period \((t+1)\). If demand is biased, then with probability 1 one of the firms (Firm 1 or another one) serves the entire demand, which leads to an expected sum of future discounted profits that is below \(W_{c-,k}\). If demand is normal and nonzero, the state of the world at the beginning of period (\(t+1)\) is ’correction at the expense of Firm 1, with k remaining periods’, leading to an expected sum of future discounted profits \(W_{c,k}\) (by Step 2). Therefore, a deviation with \(p_{t}^{1}<V\) would lead at the beginning of period t to an expected sum of future discounted profits that is less than or equal to \((V-c)D+\delta \left( \gamma ^{L}W'+\gamma ^{B}W_{c-,k}+\left( 1-\gamma ^{B}-\gamma ^{L}\right) W_{c,k}\right)\):

which implies

If \(\gamma ^{B}\) and \(\gamma ^{L}\) are close enough to zero, (3) implies that the right-hand term of this inequality is less than W, so that that \(W'<W\).

We show now that there exists no best response such that, with a positive probability, \(p_{t}^{1}>V\). If \(p_{t}^{1}>V\) and demand is neither biased nor zero, then Firm 1 earns zero in period t. In this case its deviation is detected at the end of period \((t+1)\) because a distribution of sales such that only one firm has zero sales is incompatible with equilibrium. This, and the fact that a firm’s per-period expected profit cannot exceed \((V-c)D(1+\gamma ^{B}u)\), implies that the corresponding expected sum of future discounted profits is no greater than \((V-c)D(1+\gamma ^{B}u)\left[ \left( \frac{\gamma ^{B}}{n}+\delta \right) +\gamma ^{L}\frac{\delta ^{2}}{1-\delta }\right]\). If \(\gamma ^{B}\), \(\gamma ^{L}\), and u are close enough to zero, (2) implies that this expression is less than W.

Step 5. The above steps imply that the strategy profile under consideration is an equilibrium strategy profile if \(\gamma ^{B}\), \(\gamma ^{L}\), and u are close enough to zero, and conditions (2)-(3) are satisfied. By construction, this equilibrium is symmetric, it involves only pure strategies,and the prevailing price is V in all periods along the equilibrium path with probability one. \(\square\)

Proof of Proposition 3

First, we prove that there exist triplets \((n,\delta ,k)\) such that conditions (1)-(3) jointly hold when \(A=0\) and, by continuity, for some strictly positive \(A>0\). This can be checked numerically (Table 1).

Second, we consider a specific pair (n, k) such that (1)-(3) hold for some \(\delta\) and some \(A^{*}>0\) . Holding \(A^{*}\) constant, let \(\delta {}_{min}\) and \(\delta {}_{max}\) denote the lower and upper bound of the set of values of \(\delta\) that satisfy (1)-(3) given k, n, and \(A^{*}\). Notice that \(\delta {}_{min}>0\) and \(\delta {}_{max}<1\).

We show hereafter that it is possible to construct a set of stochastic demand functions that satisfy the assumptions of the model, with \(\gamma ^{L}\), \(\gamma ^{B}\), \(\frac{\gamma ^{B}u}{\gamma ^{L}}\), and \(\frac{nu}{(1-\delta _{max})\delta _{min}^{3}\gamma ^{B}\gamma ^{L}}\) arbitrarily close to zero and such that \(Max_{Q}\frac{\mu ^{N}(Q)}{\mu ^{N}(nQ)}\), \(Max_{Q}\frac{\mu ^{N}(Q)}{\mu ^{B}(Q)}\) and \(Max_{Q}\frac{\mu ^{B}(Q)}{\mu ^{N}(Q)}\) are bounded.

We choose some \(D>0\), \(V>0\) and \(c\in (0,V)\). Let \(\varepsilon\) denote a small positive number. We define \(\gamma ^{L}(\varepsilon )=\varepsilon\) and \(\gamma ^{B}(\varepsilon )=\varepsilon ^{2}\), and we consider a stochastic demand function that is defined as follows (using the definitions of ’normal’ and ’biased’ that are introduced in Sect. 2 ): With probability \(\left( 1-\gamma ^{B}(\varepsilon )\right)\) demand is normal; otherwise it is biased. We define some \(R(\varepsilon )\), \(p(\varepsilon )\), and \(p^{B}(\varepsilon )\) such that, conditional on demand’s being normal (resp. biased), demand is drawn from a distribution with: (i) an atom in zero with probability \(\gamma ^{L}(\varepsilon )\) (resp. probability zero); (ii) the uniform distribution over \(\left( 0,R(\varepsilon )D\right)\) with probability \(1-p(\varepsilon )-\gamma ^{L}(\varepsilon )\) (resp. probability \(1-p^{B}(\varepsilon )\)); and (iii) with probability \(p(\varepsilon )\) (resp. \(p^{B}(\varepsilon )\)) the uniform distribution over \(\left( R(\varepsilon )D,nR(\varepsilon )D\right)\). For expected demand to be D both in the case of normal and biased demand, and for the probability that demand exceeds \(\frac{1}{n}\)-th of its maximum value to be no greater than \(A^{*}\) both when demand is normal and when it is biased, it is sufficient that the following equalities hold: \(\left( \left( 1-p(\varepsilon )-\gamma ^{L}(\varepsilon )\right) +(n+1)p(\varepsilon )\right) R(\varepsilon )=2\); \(\left( \left( 1-p^{B}(\varepsilon )\right) +(n+1)p^{B}(\varepsilon )\right) R(\varepsilon )=2\); and \(p(\varepsilon )=A^{*}\). These three identities yield \(R(\varepsilon )\), \(p(\varepsilon )\), and \(p^{B}(\varepsilon )\). One can check that as \(\varepsilon\) goes to zero, these distributions of demand—that are denoted \(\mu _{\varepsilon }^{N}\) (for normal demand) and \(\mu _{\varepsilon }^{B}\) (for biased demand)—are such that \(Max_{Q}\frac{\mu _{\varepsilon }^{N}(Q)}{\mu _{\varepsilon }^{N}(nQ)}\), \(Max_{Q}\frac{\mu _{\varepsilon }^{N}(Q)}{\mu _{\varepsilon }^{B}(Q)}\), and \(Max_{Q}\frac{\mu _{\varepsilon }^{B}(Q)}{\mu _{\varepsilon }^{N}(Q)}\) remain bounded: they respectively converge towards \(Max\left( \frac{1-A^{*}}{nA^{*}},1\right)\), 1, and 1 as \(\varepsilon\) converges towards zero.

Finally, we define \(u(\varepsilon )\) by the identity \(\frac{nu(\varepsilon )}{(1-\delta _{max})\delta _{min}^{3}\gamma ^{B}\gamma ^{L}(1-A^{*})}=\varepsilon\). This demand function is such that \(\gamma ^{L}\), \(\gamma ^{B}\), \(\frac{\gamma ^{B}u}{\gamma ^{L}}\), and \(\frac{nu}{(1-\delta )\delta ^{3}\gamma ^{B}\gamma ^{L}(1-A^{*})}\) are all smaller than or equal to \(\varepsilon\). Proposition 3 then implies that if there are n firms, then with any of these demand functions (if \(\varepsilon\) is small enough), the maximal attainable profits are greater with communication than without communication.

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