Silviase/jgov_v1.4_full_ann huggingface dataset:
(1) there does not exist a credible veto set for i, or

(2) there exists a credible veto set such that $ U_{i}^{k}(c|r) = 0 $ for all $ c \in V_{i} $ under a corresponding equilibrium and beliefs.

Cramton and Palfrey (1995) show that a monopoly outcome mechanism against Cournot competition in two firms and a linear demand case is not ratifiable. Tan and Yilankaya (2007) present the non-ratifiability result of bidder collusion in second-price auctions with participation cost for selling an object. We will show that the cartel mechanism against competitive first- or second-price auctions is ratifiable. The result implies that the bid-rigging is relatively stable.

## 5 Ratifiability in First-Price Auctions

To analyze ratifiability we must be cautious of asymmetry induced by belief change. Analysis of asymmetry in first-price auctions is especially difficult task and was not done until the late 1990s. Maskin and Riley (2000a) compares revenues between first- and second-price auctions. Maskin and Riley (2000b) treats the monotonicity of an equilibrium bid. Lebrun (1999) shows the existence of equilibrium. Moreover, there are few explicit solutions. Griesmer, Levitan, and Shubik (1967), Kaplan and Zamir (2007), and Tanno (2008) give explicit solutions to an auction with two bidders and uniform distribution.

In this case, we cannot use the standard technique to induce an equilibrium in symmetric auctions. We must directly solve differential equations with boundary conditions. Since it is difficult to obtain a general solution in asymmetric first-price auctions, we prove a ratifiable result in the first-price auction with two suppliers and uniform distribution. In the next section we examine differential equations and boundary conditions to solve the ratifiability problem.
### 5.1 General First-Order Condition

For the moment, we consider asymmetric bidders with a cost $ c_{i} $ that is distributed according to a distribution function $ F_{i} $ on a support $ [\underline{c}_{i}, \overline{c}_{i}] $ . Using the inverse bid function $ c_{i} = c_{i}(p_{i}) $ , general first order conditions are given by

$$ -(p_{i}-c_{i})\sum_{j\ne i}F_{j}^{\prime}c_{j}^{\prime}\Pi_{k\ne i,j}(1-F_{k})+\Pi_{j\ne i}(1-F_{j})=0\qquad for i=1,\ldots,N $$

where $ F_{j}^{\prime} $ , $ c_{j}^{\prime} $ , and $ F_{k} $ are evaluated at $ p_{i} $ , $ c_{j}(p_{i}) $ , and $ c_{k}(p_{i}) $ . Following Lebrun (1999), Maskin and Riley (2000a), and Bajari and Ye (2003), equilibrium inverse bid functions are characterized by solutions to these differential equations

$$ c_{i}^{\prime}=\frac{1-F_{i}}{(N-1)F_{i}^{\prime}}\left(\sum_{j\neq i}\frac{1}{p-c_{j}}-\frac{N-2}{p-c_{i}}\right)\qquad for i=1,\cdots,N $$

where $ c_{i} $ , $ c_{j} $ , $ F_{i} $ , and $ F_{i}^{\prime} $ are evaluated at p and $ c_{i}(p) $ over $ (\underline{p}, \overline{p}) $ satisfying some boundary conditions.

Since the belief for the vetoer is updated to some distribution function $ F_{v} $ with a veto set $ V_{v} = [l, h] $ , which is a subset of the original support, and that of the other sellers is unchanged, $ ^{12} $ the differential equations can be reduced to

$$ \begin{align*}c_{v}^{\prime}&=\frac{1-F_{v}}{(N-1)F_{v}^{\prime}}\left(\sum_{j\neq v}\frac{1}{p-c_{j}}-\frac{N-2}{p-c_{v}}\right),\\c_{i}^{\prime}&=\frac{1-F_{i}}{(N-1)F_{i}^{\prime}}\left(\sum_{j\neq i}\frac{1}{p-c_{j}}-\frac{N-2}{p-c_{i}}\right)\qquad for i\neq v.\end{align*} $$

Analytic solutions for general distributions are rare in first-price auctions. Following Kaplan and Zamir (2007) and Tanno (2008), we derive a ratifability result for a simplified auction in the case of two suppliers and uniform distribution.
### 5.2 Two bidders and Uniform Distribution Case

We consider two bidders case. Each prior cost is uniformly distributed on $ [0,1] $ . In status quo game a cost for ratifier is distributed according to $ F_{r}(c)=c $ . We consider a candidate for veto set $ V_{v}=[l,h] $ where $ 0\leq l0\quad\mathrm{f o r a l l}c. $$

Since we show that there does not exist any credible veto set and each cost type in a credible veto set is better off in the status quo game than in the cartel mechanism, it is sufficient to consider the payoff difference at the lowest reserve price.

The linear solution for the above differential equations is

$$ c_{v}(p)=2p-\frac{2+h}{3}\quad\mathrm{a n d}\quad c_{r}(p)=2p-\frac{1+2h}{3}. $$

Since this solution is a particular one, we need no boundary condition. $ ^{13} $ The price that the highest cost vetoer bids becomes the highest equilibrium
Figure 1: Two bidders and uniform distribution case in the first price auction

price $ \overline{p} $ . We assume $ r \geq \overline{p} $ by the above reason. While any ratifier who has cost higher than the cost corresponding to this highest price has zero-probability of winning, in equilibrium this type of the ratifier still follows $ c_{r} $ . By the status quo tie-breaking rule in favor of vetoer a low-cost ratifier who can set a price below the lowest price $ \underline{p} $ the vetoer bids quotes the price $ \underline{p} $ and always beats any type of the vetoer except l-type vetoer. $ ^{14} $ By taking account of conditions for high and low ends, we obtain the proposition.

Proposition 5.1 Assume that the status quo tie-breaking rule in favor of vetoer, $ l \geq 0 $ , $ h \leq 1 $ , and $ r \geq \frac{2h+1}{3} $ in the first-price auctions with uniform
distributions. The following strategies constitute equilibrium:

$$ \begin{aligned}&p_{v}=\frac{c}{2}+\frac{h+2}{6}\quad&for c\in[l,h],\\&p_{r}=\left\{\begin{aligned}&\frac{c}{2}+\frac{2h+1}{6}\quad&for c>\frac{3l-h+1}{3}\\&\underline{p}=\frac{h+3l+2}{6}\quad&for c\leq\frac{3l-h+1}{3}.\end{aligned}\right.\end{aligned} $$

See Tanno (2008) for the formal proof of the equilibrium bids. Prices and costs for each bid function are depicted in Figure 1. $ ^{15} $ Kaplan and Zamir (2007) also consider this type of solution. However, Proposition 5.1 shows the existence of a linear solution for a wider range of parameters than that in Kaplan and Zamir (2007).

The vetoer’s profit derives from Proposition 5.1:

$$ \pi_{v}^{1*}(c|r)=\frac{(h+2-3c)^{2}}{18}\qquad\mathrm{f o r}c\in[l,h]. $$

Next, we compare a collusive payoff with the status quo game payoff. According to (9), it suffices to consider the possible minimum reserve price. We substitute $ \underline{r}(h) = (2h + 1)/3 $ in the supposition of Proposition 5.1 into (8) and gets the payoff difference:

$$ \begin{align*}U_{v}^{1}(c|\underline{r}(h))&=\pi^{m}(c|\underline{r}(h))-\pi_{v}^{1*}(c|\underline{r}(h))\\&=\frac{1}{162}(8h^{3}-33h^{2}+6(7+27c)h-162c^{2}+10)\qquad for c\in[l,h].\end{align*} $$

We differentiate the above expression with respect to h:

$$ \frac{\partial U_{v}^{1}}{\partial h}(c|\underline{r}(h))=\frac{1}{27}(4h^{2}-11h+7+27c)=\frac{1}{27}\left(4\left(h-\frac{11}{8}\right)^{2}+27c-\frac{9}{16}\right). $$

We see that $ \frac{\partial U_{v}^{1}}{\partial h}(c|\underline{r}(h)) $ decreases in h for a feasible interval. The value at the maximum upper end h = 1 is

$$ \frac{\partial U_{v}^{1}}{\partial h}(c|\underline{r}(1))=c\geq0\qquad\mathrm{f o r a l l}c. $$
Therefore, we obtain

$$ \frac{\partial U_{v}^{1}}{\partial h}(c|\underline{r}(h))\geq0\qquad\mathrm{~f o r~a l l~}c\leq h\mathrm{~f o r~a l l~}h\in(0,1]. $$

Then, we can calculate the minimum of the difference of payoffs by taking limit of h and c to 0.

$$ \lim_{h,c\to0}U_{v}^{1}(c|\underline{r}(h))=\frac{5}{81}>0. $$

Therefore, we conclude that for any veto set $ V_{v} $ and reserve price r,

$$ U_{v}^{1}(c|r)>0\qquad\mathrm{f o r a n y}c\in V_{v}. $$

In this type of equilibrium, there is no credible veto set.

By following Kaplan and Zamir (2007) we analyze the payoff difference at another equilibrium. This solution is not linear and generalizes the solution that Griesmer, Levitan, and Shubik (1967) consider. The boundary condition is a little different from that for the previous linear solution. $ ^{16} $

Proposition 5.2 (Kaplan and Zamir (2007)) If the probability of winning is zero, each bidder bids his own cost. Assume that $ l \geq 0 $ , $ h \leq 1 $ , and $ r \geq \frac{2h+1}{3} $ in the first-price auctions with uniform distributions. The equilibrium inverse bid function for vetoer and ratifier is given by

$$ \begin{aligned}&c_{v}(p)=h+\frac{(1-h)^{2}}{(1+h-2p)K_{v}e^{\frac{1-h}{1+h-2p}}+4(1-p)},\\&c_{r}(p)=1+\frac{(1-h)^{2}}{(1+h-2p)K_{r}e^{-\frac{1-h}{1+h-2p}}+4(h-p)},\\ \end{aligned} $$

where

$$ K_{v}=-\frac{\frac{(1-h)^{2}}{h-l}+4(1-\underline{p})}{2(\overline{p}-\underline{p})}e^{\frac{1-h}{2(\overline{p}-\underline{p})}},\quad K_{r}=-\frac{(1-h)^{2}+4(h-\underline{p})}{2(\overline{p}-\underline{p})}e^{\frac{1-h}{2(\overline{p}-\underline{p})}}, $$

$$ \overline{p}=\frac{1+h}{2},\quad a n d\quad\underline{p}=\frac{(1+h)^{2}}{4(1+h-l)}. $$

The costs for boundaries of the bids are $ c_{v}(\overline{p}) = h $ , $ c_{v}(\underline{p}) = l $ , $ c_{r}(\underline{p}) = 0 $ , and $ c_{r}(\overline{p}) = \overline{p} $ .
See Proposition 1 in Kaplan and Zamir (2007) for the proof. While they analyze a first-price auction for selling an item, we can derive solution of an auction for buying an item in a similar way. While this equilibrium bid seems complicated, we can prove non-existence of a credible veto set in this equilibrium to focus on the payoff at the lowest vetoer's cost. It is enough to show no existence of higher payoff in the competitive auction at the lowest cost for any veto set. See Appendix B for detailed proof.

By the investigation of the payoff difference in two equilibria, we conclude that there does not exist an equilibrium and posterior belief in the status quo game in which the vetoer is better off in the status quo game than in the collusion. We prove the following proposition.

Proposition 5.3 In first-price auctions with two bidders and uniform distribution, the collusion mechanism is ratifiable against the status quo of competitive auctions.

## 6 Ratifiability in Second-Price Auctions

In this section, we show that the collusion mechanism in a second-price auction is ratifiable. Compared with a first-price auction, we treat the general setting in second-price auctions. We consider the cases for N sellers and general support. When a bidder v vetoes, other bidders update their beliefs about the vetoer in the following way: the support of belief for the vetoer is $ V = [l, h] \subset [\underline{c}, \overline{c}] $ . The updated belief is $ F_v(c) = (F(c) - F(l))/(F(h) - F(l)) $ . As we analyze the first-price auction, a bidder whose cost is above a given reserve price does not veto the collusion mechanism. We assume that the upper end is below or equal to the reserve price.

In the post-veto auction, a candidate for equilibrium is a weakly dominant strategy as is usual in second price auctions. In the case where ratifier i's cost is over h, the ratifier surely believes that he will lose to any vetoer and cannot gain from participation. Thus, choosing No is a weakly dominant strategy for this ratifier in this case. Using such a cut-off strategy, the
Figure 2: Payoff difference between collusion and post-veto auction

equilibrium $ p^{2*} = (p_{v}^{2*}, p_{i}^{2*}) $ in the post-veto auction is:

$$ p_{v}^{2*}(c)=c\mathrm{~f o r~}c\in[l,h], $$

$$ p_{i}^{2*}(c)=\left\{\begin{aligned}&No&\text{if}c\geq h\\ &c&\text{if}c
shows that if after realization of cost, a participation for bidding entails a positive entry cost for bidders, the collusion mechanism in the second-price auctions is not ratifiable against the status quo competitive auctions.

We explain the intuition behind the logic of the non-ratifiability result. A positive entry cost induces ratifiers not to take part in the auction for some costs. Let k be a threshold cost for entry. Since revenue for a winning cost-h ratifier could be at most h and his profit with participation cost would be not positive, k is strictly lower than h. Between h and k, without ratifiers, the vetoer gets more profit in competitive auction. If we choose a veto set in this area, we can show that such a veto set is credible. So, the collusion is not ratifiable. The correspondent payoff difference is depicted by the dotted line in Figure 2.

## 7 Concluding Remarks

We formulate an efficient collusion scheme in first- and second-price auctions for buying an object. In a procurement auction setting, to bring the possibility of vetoing collusion and the learning of the other bidders after vetoing, we investigate the stability of bid-rigging. We conclude that collusion in sealed low-bid auctions with uniform distribution and two bidders and general open descending auctions is stable against competition after a potential veto to take part in bid-rigging. The two leading bidding systems are not immune to collusion.

We compare our findings with related results. In a second-price auction, some participation cost hinders collusion. If we reform the bidding system to introduce some cost that influences not the production cost, but the entry decision, the new bidding system of open descending auctions is expected to reduce collusion. Collusion is easier in auctions than it is in market competition. Competition authority should be more careful with bid-rigging than price cartels. To detect and prevent collusions, procurers reform bidding systems and competition authorities use some tools, e.g., leniency programs, etc. Our conclusion partially justifies these efforts to
deter bid-riggings.

In future research we will investigate a bidding system in which a cartel is not ratifiable. To formulate better competition policy, it is important to investigate what conditions or institutional characteristics affect collusion.

### A Proof of Lemma 3.2

Proof: The collusive payoff for a firm with cost c is given by

$$ \begin{align*}\pi^{m}(c|r)&=(r-c-T(c))(1-F(c))^{N-1}+\int_{\underline{c}}^{c}T(x)(1-F(x))^{N-2}f(x)dx\\&=(r-c)(1-F(c))^{N-1}\\&-(N-1)(1-F(c))^{-1}\int_{c}^{r}(r-x)(1-F(x))^{N-1}f(x)dx\\&+(N-1)\int_{\underline{c}}^{c}(1-F(x))^{-2}f(x)\int_{x}^{r}(r-y)(1-F(y))^{N-1}dydx.\end{align*} $$

Note that by integral by parts,

$$ \int_{c}^{r}(r-x)(1-F(x))^{N-1}f(x)dx=\frac{1}{N}\left((r-c)(1-F(c))^{N}-\int_{c}^{r}(1-F(x))^{N}dx\right). $$

Applying (12) and a kind of (12) to the second and the third terms in (11), we get

$$ \begin{align*}\pi^{m}(c|r)&=\frac{1}{N}(r-c)(1-F(c))^{N-1}+\frac{N-1}{N}(1-F(c))^{-1}\int_{c}^{r}(1-F(x))^{N} dx\\&+\frac{N-1}{N}\int_{\underline{c}}^{c}(r-x)(1-F(x))^{N-2}f(x)dx\\&-\frac{N-1}{N}\int_{\underline{c}}^{c}(1-F(x))^{-2}f(x)\int_{x}^{r}(r-y)(1-F(y))^{N-1}dy dx.\ (13)\end{align*} $$
We obtain that by changing the order of integration,

$$ \begin{align*}&\int_{\underline{c}}^{c}\frac{\int_{x}^{r}(1-F(y))^{N} dy}{(1-F(x))^{2}}f(x)dx\\=&\int_{c}^{r}\int_{\underline{c}}^{c}\frac{(1-F(y))^{N}}{(1-F(x))^{2}}f(x)dx dy+\int_{\underline{c}}^{c}\int_{\underline{c}}^{y}\frac{(1-F(y))^{N}}{(1-F(x))^{2}}f(x)dx dy\\=&\int_{c}^{r}(1-F(y))^{N}((1-F(c))^{-1}-1)dy+\int_{\underline{c}}^{c}(1-F(y))^{N}((1-F(y))^{-1}-1)dy\\=&\int_{c}^{r}\frac{(1-F(x))^{N}}{1-F(c)}dx+\int_{\underline{c}}^{c}(1-F(x))^{N-1}dx-\int_{\underline{c}}^{r}(1-F(x))^{N} dx.\end{align*} $$

Using this equations and a kind of (12), (13) is equal to

$$ \begin{align*}\pi^{m}(c|r)=&\frac{1}{N}(r-\underline{c})-\int_{\underline{c}}^{c}(1-F(x))^{N-1}dx+\frac{N-1}{N}\int_{\underline{c}}^{r}(1-F(x))^{N} dx\\=&\frac{1}{N}(r-\underline{c})+\frac{N-1}{N}\int_{\underline{c}}^{r}(1-F(x))^{N} dx-\int_{\underline{c}}^{r}(1-F(x))^{N-1}dx\\&+\int_{c}^{r}(1-F(x))^{N-1}dx.\end{align*} $$

Especially,

$$ \pi^{m}(r|r)=\frac{1}{N}(r-\underline{c})+\frac{N-1}{N}\int_{c}^{r}(1-F(x))^{N}d x-\int_{c}^{r}(1-F(x))^{N-1}d x. $$

Next, we transform this fixed term. We note that

$$ \begin{align*}&((r-x)(1-F(x))^{N-1}F(x))^{\prime}=-(1-F(x))^{N-1}F(x)\tag{14}\\&\quad-(r-x)(N-1)(1-F(x))^{N-2}F(x)f(x)+(r-x)(1-F(x))^{N-1}f(x).\end{align*} $$

Rearranging $ \pi^{m}(r|r) $ gives

$$ \begin{align*}\pi^{m}(r|r)=\frac{1}{N}(r-\underline{c})-\int_{\underline{c}}^{r}(1-F(x))^{N-1}F(x)dx-\frac{1}{N}\int_{\underline{c}}^{r}(1-F(x))^{N} dx\\=-\int_{\underline{c}}^{r}(1-F(x))^{N-1}F(x)dx+[-\frac{1}{N}(r-x)(1-F(x))^{N}]_{\underline{c}}^{r}-\frac{1}{N}\int_{\underline{c}}^{r}(1-F(x))^{N} dx\\=-\int_{\underline{c}}^{r}(1-F(x))^{N-1}F(x)dx+\int_{\underline{c}}^{r}(r-x)(1-F(x))^{N-1}f(x)dx\\=[-(r-x)(1-F(x))^{N-1}F(x)]_{\underline{c}}^{r}\\-\int_{\underline{c}}^{r}(1-F(x))^{N-1}F(x)dx+\int_{\underline{c}}^{r}(r-x)(1-F(x))^{N-1}f(x)dx\\=\int_{c}^{r}(r-x)(N-1)(1-F(x))^{N-2}F(x)f(x)dx.\end{align*} $$

The last equality uses (14).
### B Proof of Proposition 5.3

If for some $ V_{v} $ we obtain $ U_{v}^{1}(c|r) < 0 $ for all $ c \in V_{v} $ , it is also true that $ U_{v}^{1}(l|r) < 0 $ . In order to prove no credibility, we focus on the lowest cost and prove $ U_{v}^{1}(l|r) > 0 $ for any $ V_{v} $ and r. By (10), the competitive payoff for the vetoer at the lowest cost is given by

$$ \pi_{v}^{1*}(l|r)=(1-F_{r}(c_{r}(\underline{p})))(\underline{p}-c_{v}(\underline{p}))=\frac{(1+h-2l)^{2}}{4(1+h-l)}. $$

The supposition of Proposition (5.2), the possible minimum reserve price is $ \underline{r} = (1 + h)/2 $ .

By the above reasoning, substituting $ \underline{r} $ into (8) yields the payoff difference:

$$ \begin{align*}U_{v}^{1}(l|\underline{r})&=\pi^{m}(l|\underline{r})-\pi_{v}^{1*}(l|\underline{r})\\&=\frac{h^{4}-(2+l)h^{3}+27lh^{2}+(-48l^{2}+33l+10)h+24l^{3}-48l^{2}+5l+7}{48(1+h-l)}.\end{align*} $$

Note that the denominator in the payoff difference is always positive for feasible valuables. Let $ \zeta(h;l) $ be the numerator and $ \eta(l) $ be $ \lim_{h\to l}\zeta(h;l)=l^{3}-15l^{2}+15l+7 $ . We will prove that $ \zeta(h;l) $ is increasing in h for $ l0 $ for any l.

The derivatives of $ \zeta(h;l) $ are

$$ \begin{align*}\frac{d\zeta}{dh}(h;l)&=4h^{3}-3(2+l)h^{2}+54lh-48l^{2}+33l+10,\\\frac{d^{2}\zeta}{dh^{2}}(h;l)&=12h^{2}-6(2+l)h+54l.\end{align*} $$

We easily see that $ \lim_{h\to l}\frac{d\zeta}{dh}(h;l)=l^{3}+33l+10>0 $ . We denote the discriminant of the quadratic equation $ \frac{d^{2}\zeta}{dh^{2}}(h;l)=0 $ by $ d(l)=l^{2}-68l+4 $ . Let $ l^{*} $ be the smaller solution to $ d(l)=0 $ . By simple calculation we obtain $ l^{*}=34-24\sqrt{2}\simeq0.059 $ . First, we easily see that when the lower bound is large enough for $ l>l^{*} $ in the case of $ d(l)<0 $ , $ \frac{d^{2}\zeta}{dh^{2}}(h;l) $ is positive for all h.

Second, when l is lower than $ l^{*} $ , the first order derivative $ \frac{d\zeta}{dh}(h;l) $ is decreasing in h in the interval between $ (2 + l - \sqrt{d(l)})/4 $ and $ h^{*}(l) := $
$ (2 + l + \sqrt{d(l)})/4 $ . Therefore, in this case the first order derivative attains a local minimum at $ h^{*}(l) $ . Substituting $ h^{*}(l) $ in it yields:

$$ \frac{d\zeta}{d h}(h^{*}(l);l)=\frac{1}{8}(-l^{3}-282l^{2}+468l+72-d(l)^{\frac{3}{2}}). $$

We easily see that $ \frac{d\zeta}{dh}(h^{*}(0);0)=8>0 $ . Since $ d(l) $ is decreasing in $ l\leq l^{*} $ , we obtain that $ -d(l)^{\frac{3}{2}} $ is increasing in the region. Let $ \theta(l) $ be $ -l^{3}-282l^{2}+468l $ . Each solution of the equation $ \theta'(l)=-3l^{2}-564l+468=0 $ is $ -94\pm4\sqrt{562} $ . We easily compute that the large one ( $ \simeq0.83 $ ) is greater than $ l^{*} $ and the small one is negative. We obtain that $ \theta'(l)>0 $ and then $ \theta(l)\geq0 $ in the region. Therefore, we conclude $ \frac{d\zeta}{dh}(h^{*}(l);l)>0 $ in the region.

Finally, to bring two cases together we obtain that $ \frac{d\zeta}{dh}(h;l)>0 $ for any feasible h. We prove that $ \zeta(h;l) $ is positive for any h and l with $ 0\leq l0\quad for c 0 $ . Hence, there exists h such that $ g(h) = 0 $ for h < r.
Let the vetoer v's payoff after a veto be $ \pi_{v}^{2*} $ . We will show that $ \pi_{v}^{2*}(c) \leq \pi^{m}(c) $ for any c. As in (2), the equilibrium strategy gives

$$ \pi_{v}^{2*}(c)=\left\{\begin{aligned}&0&&if c>r,\\&(r-c)(1-F(h))^{N-1}&&if r\geq c>h,\\&(r-c)(1-F(h))^{N-1}&&\\&\quad+\int_{c}^{h}(x-c)(N-1)(1-F(x))^{N-2}f(x)dx&&if h\geq c.\end{aligned}\right. $$

We denote $ \pi^{m}(c)-\pi_{v}^{2*}(c) $ by $ U_{v}^{2}(c) $ . We obtain that by (4) and (5),

$$ U_{v}^{2}(c)=\left\{\begin{aligned}&\pi^{m}(r|r)&&if c>r,\\&\pi^{m}(r|r)+\int_{c}^{r}(1-F(x))^{N-1}dx-(r-c)(1-F(h))^{N-1}&&if r\geq c>h,\\&\pi^{m}(r|r)+\int_{c}^{r}(1-F(x))^{N-1}dx-(r-c)(1-F(h))^{N-1}\\&&-\int_{c}^{h}(x-c)(N-1)(1-F(x))^{N-1}f(x)dx&&if h\geq c.\end{aligned}\right. $$

First, we consider the case for $ r \geq c > h $ :

$$ U_{v}^{2}(c)=\pi^{m}(r|r)+\int_{c}^{r}(1-F(x))^{N-1}d x-(r-c)(1-F(h))^{N-1}. $$

Then,

$$ U_{v}^{2^{\prime}}(c)=\begin{cases}-(1-F(c))^{N-1}+(1-F(h))^{N-1}>0&\text{if}c>h\\0&\text{if}c=h.\end{cases} $$

Since $ U_{v}^{2}(h)=0 $ by (15), we conclude that $ U_{v}^{2}(c)\geq0 $ for $ r\geq c\geq h $ .

Second, we consider the case for c < h.

$$ \begin{align*}U_{v}^{\prime}(c)&=-(1-F(c))^{N-1}+(1-F(h))^{N-1}+\int_{c}^{h}(N-1)(1-F(x))^{N-2}f(x)dx\\&=-(1-F(c))^{N-1}+(1-F(h))^{N-1}-[(1-F(c))^{N-1}]_{c}^{h}=0.\end{align*} $$

Since $ U_{v}^{2}(h)=0 $ , we see that $ U_{v}^{2}(c)=0 $ for $ c\leq h $ . The difference of payoffs corresponds to the bold line in Figure 2. Even if the vetoer makes the ratifiers stay out, he only gets his payoff equal to the bid-rigging payoff in this case. Since the vetoer does not strictly gain by vetoing in any case, the candidate veto set is not credible. Since the equilibrium $ p^{*} $ is a weakly
dominant strategy, there is no credible belief and no strategy pertaining to it.

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