Optimal implicit collusion in repeated procurement auctions

Abstract

We develop a model of implicit collusion in repeated procurement auctions in which suppliers can only observe past auction prices, but not all bids and the identities of winners. We focus on symmetric perfect public equilibria (SPPE) and use the dynamic programming techniques to characterize the optimal SPPE. We allow for a public randomization device and find that the implementation of the optimal collusive equilibrium can be simply obtained by using the bang-bang property, because suppliers just need to adopt the collusive bidding schedule and depending on the past winning bids, with a certain probability they stick to that strategy or move to Nash reversion. The optimal implicit collusion is characterized by a rigid-bidding scheme with punishments occurring on the equilibrium path. As a result, the optimal collusion can not achieve full efficiency.

Appendices

Appendix A: Proof of Proposition 1

For each $W\subseteq \mathbb {R}$, define $B(W)=\{E\{\Pi (c;\beta ,u)\}|(\beta ,u) ~is~ admissible~ with respect~ to~ W\}$. $B(W)\subseteq R$ represent the collusive suppliers’ total expected payoffs in pure strategy equilibrium in which each first-period stage game gain is followed by some symmetric expected future payoff drawn from W. At the end of the single period of the truncated game, suppliers receive their conventional one-period profits, plus some element of W (the same for each supplier), depending on the past winning bids that arises; the expected value of this sum, discounted to the beginning of the period, will be the one element of B(W).

For any SPPE s, the value to supplier i of the successor SPPE specified following a given first-period winning price $b_w$ must be independent of the bid that i submitted in period 1. This is because no one else has observed that bid, and hence i faces the same future environment regardless of his initial bid. Thus the value of s for each supplier can be factorized into two terms: the gain from first-period game, and the discounted expected value of a reward function $E_{b_w}(u(b_w)|\beta )=v(s|_{h(1)=(b_w^1,s^1)})$. Since $s|_{h(1)=(b_w^1,s^1)}$ is an SPPE, this reward function is drawn from V. This together with the constraints that the suppliers are willing to adopt strategy $s^1=(\beta _1,\cdots ,\beta _n)$ in period 1, means precisely that $(\beta ,u)$ is admissible with respect to V. Thus the requirements for s to be an SPPE are exactly those needed for v(s) to be in B(V), and therefore $V=B(V)$.

We have shown that V can be recovered from the function B. In addition, V is a fixed point of B and V is also the largest bounded fixed point of B. Moreover, since V is self-generating, any element w of V is the payoff of $\hat{s}(w)$, the SPPE constructed in Proposition 1.¹⁵ This supergame equilibrium is described entirely by two functions $\beta$ and u, and the number w.

Appendix B: Proof of Proposition 2

Gains to a player with cost parameter c from the collusive mechanism $(\beta ,u)$ are:

$$\begin{aligned} \int _c^{\overline{c}}Q(x;\beta )dx+\delta \int _{\underline{c}}^{\overline{c}}u(\beta (c))(1-F(c))^{n-1}dF(c) \end{aligned}$$

which can be used to calculate ex-ante rents:

$$\begin{aligned} \int _{\underline{c}}^{\overline{c}}H(c)Q(c;\beta )dF(c)+\delta \int _{\underline{c}}^{\overline{c}}u(\beta (c))(1-F(c))^{n-1}dF(c) \end{aligned}$$

Consider infinite, or unrelenting punishments. Let $(\beta _1,U_1)$ be an incentive compatible collusive mechanism which minimizes the above equation:

$$\begin{aligned} \min _{\beta _1,U_1}:\int _{\underline{c}}^{\overline{c}}H(c)Q(c;\beta )dF(c)+\delta \int _{\underline{c}}^{\overline{c}}u(\beta (c))(1-F(c))^{n-1}dF(c) \end{aligned}$$

Since payoffs are constrained to be in V, the perfect equilibrium set, and we have proved in Proposition 2 that $V=B(V)$, there exists another collusive couple, $(\beta _2,u_2)$ giving the same payoffs as $\int _{\underline{c}}^{\overline{c}}\Big [E_b(u_1(b)|\beta _1(c))\Big ]dF(c)$. Working iteratively in this manner one obtains that the payoff to this punishment is

$$\begin{aligned} \sum _{i=1}^\infty \delta ^{i-1}\int _{\underline{c}}^{\overline{c}}H(v)Q(c;\beta _i)dF(v) \end{aligned}$$

Incentive compatibility requires $Q(\cdot ;\beta )$ to be non decreasing. The above expression is minimized when we rule out constant bidding regions, i.e., when the bidding rules are constrained to induce efficiency so that $Q(c;\beta )=(1-F(c))^{n-1}$. The single stage Nash strategy accomplishes this task while satisfying incentive compatibility. Note that when bidding rules are constrained to induce efficiency payoffs are necessarily equal to the Nash equilibrium stage game payoffs.

The next step is to show that participants always bid with probability one. Suppose a mechanism which randomly selects agents not to bid. The identity of the winning bidder is unobservable, so future gains can only be a function of the winning bid. Therefore for a certain agent with a given type to be indifferent between bidding and not bidding, all agents with lower cost must strictly prefer bidding and all agents with higher cost must strictly prefer not bidding. Consider a mechanism that in the first round outlaws bidding above $r>0$, and let $\mu _0$ be the future expected gains in nobody bids. Admissibility of the mechanism implies:

$$\begin{aligned} \delta \mu _0(1-F(r))^{n-1}\ge r(1-F(r))^{n-1}+\delta \underline{u}(1-F(r))^{n-1} \end{aligned}$$

And since $(1-F(r))^{n-1}>F(x)(1-F(x))^{n-1}$ for all x in [0, r] we have that:

$$\begin{aligned} r(1{-}F(r))^{n-1}{+}\delta \underline{u}(1-F(r))^{n-1}{>}\int _0^rF(x)(1-F(x))^{n-1}dx{+}\delta \underline{u}(1-F(r))^{n-1} \end{aligned}$$

This previous expression represents using the Nash bidding strategy in the interval [0, r]. Therefore we have contradicted the supposition that outlawing bidding in a certain region dominated the Nash bidding strategy.

Appendix C: Proof of Proposition 3

Let $\{Q(\cdot ;\beta ),u(\cdot )\}$ be an optimal collusive mechanism, generating $\overline{u}$, for which the bang-bang property does not hold. Therefore, there exists an interval, $[c_1,c_2]$, such that $u(\beta (c))<\overline{u}$ for all $c\in [c_1,c_2]$. In this case we show that there exists an alternative incentive compatible, admissible mechanism which generates higher rents for all types. This mechanism consists of raising $u(\beta (c))$ by some small amount $\Delta u$ whenever $c\in [c_1,c_2]$. Denote this new continuation function by $u^*$. All flating bidding regions above $c_2$ are cut into two regions otherwise, the probability of winning is held constant for all other types. From equation (1), if we increase continuation payoffs in $[c_1,c_2]$ and wish to preserve type conditional probabilities of winning an auction ($Q(\cdot ,\beta )$), it is necessary to change bids in order to retain incentive compatibility. Bids of types lower than $c_1$ are unaffected by such a change. Studying inequality (2) immediately shows that types in $[c_1,c_2]$ will satisfy admissibility. The variation in the rents to $c<c_2$ are given by:

$$\begin{aligned} \Delta U(c_2)=[(1-F(c_1))^{n-1}-(1-F(c_2))^{n-1}]\Delta u \end{aligned}$$

which is strictly positive for all $\Delta u>0$.

Consider the interval $[\underline{c}_i,\overline{c}_i]$ where $\underline{c}_i\ge c_2$ and for which the bidding function is flat. Find recursively a $\hat{c_i}$ where the following holds:

$$\begin{aligned} (\hat{c}_i-\underline{c}_i)[Q(\underline{c}_i,\overline{c}_i)-Q(\underline{c}_i,\hat{c}_i)]=\Delta U(\underline{c}_i) \end{aligned}$$

Create a new incentive compatible bidding function using the new continuation function $u^*$, $\beta ^*$ different from $\beta$ only in that $Q(\cdot ;\beta ^*)$ exhibits discontinuities at all $\hat{c}_i$ but constant on all the $[\underline{c}_i,\hat{c}_i]$ and $[\hat{c}_i,\overline{c}_i]$.

We now have $\Delta U(c)=(c-\hat{c}_i)[Q(\hat{c}_i,\overline{c}_i-Q(\underline{c}_i,\overline{c}_i)]\ge 0$ for all $c\in [\hat{c}_i,\overline{c}_i]$. Additionally for all $c\in [\overline{c}_i,\hat{c}_i]$, we have $\Delta U(c)=[c-\underline{c}_i][Q(\underline{c}_i,\hat{c_i})-Q(\underline{c}_i,\overline{c}_i)]+\Delta U(\underline{c}_i)\ge 0$. Hence the variation in rents for all types is non negative by construction, which implies that this new constructed mechanism can generate higher rents for all types. It remains to verify that the new mechanism is admissible for all $c>c_2$ if the original mechanism was assumed to be admissible.

First consider checking admissibility in $[\hat{c}_i,\overline{c}_i]$. With the original mechanism we have

$$\begin{aligned} \beta (\overline{c}_i)-\overline{c}_i=(\overline{c}_i-\hat{c}_i)+\frac{U(\hat{c}_i;\beta ,u)}{Q(\underline{c}_i,\overline{c}_i)} \end{aligned}$$

Under the new mechanism we have that:

$$\begin{aligned} \beta ^*(\overline{c}_i)-\overline{c}_i=(\overline{c}_i-\hat{c}_i)+\frac{U(\hat{c}_i;\beta ^*,u^*)}{Q(\hat{c}_i,\overline{c}_i)} \end{aligned}$$

Since $U(\hat{c}_i;\beta ,u)=U(\hat{c}_i;\beta ^*,u^*)$, and $Q(\underline{c}_i,\overline{c}_i)<Q(\hat{c}_i,\overline{c}_i)$ we have that:

$$\begin{aligned} \beta ^*(\overline{c}_i)-\overline{c}_i<\beta (\overline{c}_i)-\overline{c}_i \end{aligned}$$

Since the original mechanism was assumed to be admissible, i.e.,

$$\begin{aligned} \delta [\overline{u}-\underline{u}][1-F(\overline{c}_i)]^{n-1}\ge [\beta (\overline{c}_i)-\overline{c}_i][(1-F(\overline{c}_i))^{n-1}-Q(\overline{c}_i;\beta ,u)] \end{aligned}$$

Also we know that $Q(\overline{c}_i;\beta ^*)>Q(\overline{c}_i;\beta ,u)$ and $\beta ^*(\overline{c}_i)-\overline{c}_i<\beta (\overline{c}_i)-\overline{c}_i$, and thus

$$\begin{aligned}&[\beta (\overline{c}_i)-\overline{c}_i][(1-F(\overline{c}_i))^{n-1}-Q(\overline{c}_i;\beta ,u)]\\&\quad >[\beta ^*(\overline{c}_i)-\overline{c}_i][(1-F(\overline{c}_i))^{n-1}-Q(\overline{c}_i;\beta ^*,u^*)] \end{aligned}$$

So the admissibility constraint of the new mechanism is sure to be satisfied. Finally, check admissibility in $[\underline{c}_i,\hat{c}_i]$. Note that as $\Delta u$ approaches zero, $\hat{c}_i$ must approach $\underline{c}_i$. It follows that for a $\Delta u$ sufficiently close to zero, the admissibility constraint will be respected for $\hat{c}_i$.

Appendix D: Proof of Proposition 4

Proof

To find the most collusive SPPE, I have recured to the recursive dynamic programming technique developed by Abreu et al. (1986, 1990). Any SPPE of the repeated game can be decomposed into a pair of first period price schedules $\beta (\cdot )$ and a continuation value function $u(\cdot )$. The continuation payoff $u(\cdot )$ depends on the public history of the first period, i.e. winning bid. In order for a decomposition pair $(\beta (\cdot ),u(\cdot ))$ to qualify as a SPPE, two necessary and sufficient conditions have to hold. First, an individual supplier should have no incentive to deviate from the current period price schedule given all other firms choose $\beta (\cdot )$. And second, all continuation values are drawn from the set of SPPE payoff values V. These two conditions are given by Eq.(4) and (5). Since the perpetual repetition of the Nash equilibrium of the stage game is always a SPPE, the set V is non-empty and its lowest possible value is $\underline{u}$. Convexify the set V by assuming that suppliers have access to a public randomization device at the end of each period. Then, the bang-bang property of optimal continuation values in an equilibrium implies that the value of any SPPE (including the most collusive) can be sustained by a SPPE which after every public history of the first period only uses the two extreme continuation values $\overline{u}$ and $\underline{u}$. In other words, the choice of optimal continuation values can be reduced to assigning a probability $\lambda \in (0,1)$ to every possible history after period 1. With probability $\lambda$ suppliers revert to the stage game Nash equilibrium forever. With the remaining probability suppliers continue to play the current period strategy and receive a continuation value of $\overline{u}$. $\square$

Appendix E: Proof of Lemma 1

Suppose $\beta$ to be constant on $[c_i,c_j]$. First remark that $Q(c,\beta )$ is discontinuous at $c_i$, and that in particular $Q(c_i,\beta )<[1-F(c_i)]^{n-1}$. Suppose that the condition in the statement of the lemma is violated and that $\beta$ is continous at $c_i$. Then since $u(\beta (c_j-\varepsilon ))\ge \underline{u}$ for all $\varepsilon >0$, there exists a small $\varepsilon$ such that bidders of type $c_i$ prefer to bid $\beta (c_j-\varepsilon )$, violating the incentive compatibility constraints. Now suppose that $\beta$ is discontinuous at $c_i$. From equation (1), one can verify that $\beta$ is also discontinuous at $c_j$. Therefore there exists a $b\notin \beta$ which is arbitrarily close to $\beta ([c_i,c_j])$ yet which discretely increases the probability of winning. A (credible) punishment must be available to dissuade such a deviation. This punishment can be made as severe as possible without harming cartel gains, in so far as it is only used off the equilibrium path, i.e., $b_w\notin \beta$.

To prove the only if part, notice that admissibility implies that all types respect the above inequality, and in particular $c_i$ respect the above inequality.

To prove the if part, define $\beta ^-=\lim _{c\rightarrow c_i^-}\beta (c)$ and $\beta ^+=\lim _{c\rightarrow c_i^+}\beta (c)$. Since $\beta$ is discontinuous at $c_i$, we have $\beta ^+>\beta ^-$. Any bid in $(\beta ^-,\beta ^+]$ will win the auction with probability $(1-F(c_i))^{n-1}$. The inequality in the statement of the lemma implies that $c_i$ prefers following his equilibrium strategy rather than deviating. Fix $c>c_i$. By monotonicity of preferences, c also prefers bidding $\beta ^-$ than bidding in $(\beta ^-,\beta ^+]$. A similar argument shows that all $c<c_i$ prefer bidding $\beta ^-$ than bidding in $(\beta ^-,\beta ^+]$. $\quad \square$

Appendix F: Proof of Lemma 2

Suppose an optimal collusive bidding function which is continuously increasing for some interval $[c_0,c_1]\subset [\underline{c},\overline{c}]$. To show a contradiction, modify the bidding rule so that it awards the object to types in $[c_0,c_1]$ with equal probability and respects incentive compatibility. There are two cases which will be considered separately: (1) type $c_0$ weakly prefers the old scheme. (2) Type $c_0$ strictly prefers the new scheme. In each treatment incentive compatibility is verified while using the original value $\overline{u}$, then it is shown that the new bidding function generates a higher $\overline{u}$.

Case 1  Suppose that after invoking $b_0'$, type $c_0$ is just as well off as under the old bidding function. Therefore bidding levels $b_1$ and $b_2$ can be maintained and all incentive compatibility constraints are satisfied. Now apply lemma 2 to see that the expected value to collusion has been increased. Suppose that type $c_0$ strictly prefers the old scheme. Then if $b_1$ and $b_2$ are held constant, the type indifferent between bidding $b_0'$ and $b_1$ will be located to the right of $c_0$. Now raise $b_1$ and $b_2$ such that the indifferent type between bidding $b_0'$ and $b_1$ remains $c_2$. Notice that this change in the bidding function has not changed the probabilities that any type lower than $c_1$ wins the auction and all incentive compatibility and admissibility constraints are respected. Now apply Lemma 2.

Case 2  Suppose that type $c_0$ strictly prefers the new scheme. Then the type indifferent between bidding $b_0'$ and $b_1$ moves to the left of $c_1$. Call this new indifferent type $c_1'$. Incite $c_1'$ to move back to $c_1$ by decreasing the continuation payoff if the winning bid is $b_0'$. Such a change produces a more profitable bidding function be Lemma 2, but may not produce higher rents. Invoke Proposition 4 to be assured of the existence of a collusive mechanism with a bidding function which is never continuously increasing and having the bang-bang property. This bang-bang mechanism uses a bidding function which is even more profitable. Furthermore, such a bang-bang mechanism generates $\overline{u}=\int _{\underline{c}}^{\overline{c}}H(c)Q(c;\beta ^*)dF(c)$, where $\beta ^*$ is the newest bidding function. So since the bang-bang mechanism produces rents which are generated by a bidding function which dominates the original scheme, the proposition is proved. $\quad \square$

Appendix G: Proof of Lemma 3

The proof of Lemma 2 makes reference to the following lemma which we state and prove before the proof of Lemma 2.

Lemma 3

For any $\underline{c}<c_0<c_1<\overline{c}$ , it is the case that:

$$\begin{aligned} \frac{\int _{c_0}^{c_1}H(c)dF(c)}{F(c_1)-F(c_0)}\ge \frac{\int _{c_0}^{c_1}H(c)dF(c)^n}{(1-F(c_0))^n-(1-F(c_1))^{n}} \end{aligned}$$

Proof of Lemma 3. After an integration by parts the above inequality can be written as:

$$\begin{aligned} \int _{c_0}^{c_1}\frac{F(c_1)-F(c)}{F(c_1)-F(c_0)}H'(c)dc\ge \int _{c_0}^{c_1}\frac{(1-F(c))^n-(1-F(c_1))^{n}}{(1-F(c_0))^n-(1-F(c_1))^{n}}H'(c)dc \end{aligned}$$

According to Assumption (A2), we know that $H'(c)>0$, it suffices to show that:

$$\begin{aligned} \frac{(1-F(c))-(1-F(c_1))}{(1--F(c_0))-(1-F(c_1))}\ge \frac{(1-F(c))^n-(1-F(c_1))^{n}}{(1-F(c_0))^n-(1-F(c_1))^{n}},~\forall c\in (c_0,c_1) \end{aligned}$$

Since it is the case that$F(c_0)<F(c)<F(c_1)$, $1-F(c)$ can be written as:

$$\begin{aligned} 1-F(c)=\lambda (1-F(c_0))+(1-\lambda )(1-F(c_1))~~for~some~ \lambda \in (0,1) \end{aligned}$$

Therefore after substitution and rearrangement it is sufficient to show that:

$$\begin{aligned} \Big [\lambda (1-F(c_0))+(1-\lambda )(1-F(c_1))\Big ]^n\le \lambda (1-F(c_0))^n+(1-\lambda )(1-F(c_1))^n \end{aligned}$$

which is always satisfied since $(1-F(c))^n$ is a convex function of $(1-F(c))$. $\square$