Entry Deterrence and Free Riding in License Auctions: Incumbent Heterogeneity and Monotonicity

Case 1.

Consider b _−i such that there is an incumbent other than i who bids at least w _e , that is, max { b j : j ≠ i } ≥ w e . Suppose b i ≥ max   { b j : j ≠ i } . If i wins, then u i ( b i , b − i ) = w i − max { b j : j ≠ i } ≤ w i − w e ; otherwise, u i ( b i , b − i ) = π i ( n ) . If i bids b ′ i < w e , then, since another incumbent deters the entry given b − i , i’s payoff is π i ( n ) . Since w i − w e ≤ π i ( n + 1 ) ≤ π i ( n ) (the first inequality holds for i ∈ I \ H ) and i has a positive chance of winning given ( b i , b − i ) , the payoff from bidding b i ′ , π i ( n ) , is greater than or equal to the payoff from bidding b i , which is a “strict mixture” of w i − max { b j : j ≠ i } ( ≤ w i − w e ) and π i ( n ) . Suppose b i < max { b j : j ≠ i } . Then, i’s payoff is π i ( n ) , which is the same as the payoff from bidding b ′ i < w e .

Case 2.

Consider b − i such that all other incumbents bid less than w e , that is, max { b j : j ≠ i } < w e . Then, u i ( b i , b − i ) = w i − w e . If i bids b i ′ < w e , an entrant wins the license, and u i ( b ′ i , b − i ) = π i ( n + 1 ) . Since i ∈ I \ H , w i − w e ≤ π i ( n + 1 ) , so bidding b ′ i yields a payoff that is greater than or equal to the payoff from bidding b i . □

Proof of Proposition 1 We first show that any b ∈ ℝ + n \ ∪ i ∈ H B i is not an equilibrium. By Lemma 1, for i ∈ I \ H , bidding b i ≥ w e is weakly dominated. Let each i ∈ H bid b i < w e . Then for j ∈ H , u j ( b j , b − j ) = π j ( n + 1 ) , but by bidding b ′ j ≥ w e , u j ( b ′ j , b − j ) = w j − w e . Since w e < v j + Δ π j , u j ( b ′ j , b − j ) > u j ( b j , b − j ) . Let i , j ∈ H bid b i ≥ w e and b j ≥ w e , respectively. Then u i ( b i , b − i ) ≤ w i − w e , but by bidding b i ′ < w e , u i ( b i ′ , b − i ) = π i ( n ) . Since w e > max i ∈ I v i , u i ( b i ′ , b − i ) > u i ( b i , b − i ) . Next, we show that for all i ∈ H , each b ∈ B i is a pure strategy Nash equilibrium. Let i ∈ H and b ∈ B i . Then u i ( b i , b − i ) = w i − w e > π i ( n + 1 ) = u i ( b i ′ , b − i ) for any b i ′ < w e since w e < v i + Δ π i , and u i ( b i , b − i ) = w i − w e = u i ( b i ′ , b − i ) for any b i ′ ≥ w e with b i ′ ≠ b i . Let j ≠ i . Then u j ( b j , b − j ) = π j ( n ) > w j − w e ≥ u j ( b j ′ , b − j ) for any b j ′ ≥ b i since w e > max i ∈ I v i , and u j ( b j , b − j ) = π j ( n ) = u j ( b j ′ , b − j ) for any b j ′ < b i . □

Proof of Lemma 2 Suppose by contradiction that Pr σ i [ b i > w e ] > 0 . Then, either (i) for some h ∈ H , Pr σ [ b h > max j ∈ H \ { h } b j ≥ w e ] > 0 , or (ii) there is b ˆ > w e such that for each i ∈ H participating in preemption, Pr σ i [ b i = b ˆ ] + Pr σ i [ b i < w e ] = 1 (that is, each participating incumbent in H chooses the same deterring bid b ˆ above w e or a losing bid below w e ).

Case 1.

For some h ∈ H , Pr σ [ b h > max j ∈ H \ { h } b j ≥ w e ] > 0 . Let σ h ′ be a mixed strategy that is obtained by lowering all bids higher than w e given σ h to w e , i.e.,  Pr σ h ′ [ b h = w e ] = Pr σ h [ b h ≥ w e ] and for all b ′ h < w e , Pr σ h ′ [ b h ≤ b h ′ ] = Pr σ h [ b h ≤ b h ′ ] . Let B * ≡ { b : b h > max j ∈ H \ { h } b j ≥ w e } . For each b ∈ B * , by switching from σ h to σ h ′ , h’s payoff strictly increases from u h ( b ) ≤ w h − w e to π h ( n ) or α ( w h − w e ) + ( 1 − α ) π h ( n ) for some α = 1 2 , 1 3 , … , 1 n − 1 . For each b ∈ B \ B * , by switching from σ h to σ h ′ , h’s payoff either increases or remains the same. Since Pr σ [ B * ] > 0 , h’s expected payoff from σ h ′ is greater than h’s expected payoff from σ h , contradicting σ being a Nash equilibrium.

Case 2.

There is b ˆ > w e such that for each i ∈ H participating in preemption, Pr σ i [ b i = b ˆ ] + Pr σ i [ b i < w e ] = 1 . Pick any h ∈ H with Pr σ h [ b h = b ˆ ] > 0 . Define σ h ′ as in Case 1. Let B ˆ ≡ { b : b h = b ˆ = max j ∈ H \ { h } b j } . Then, after switching from σ h to σ h ′ , for any b ∈ B ˆ , h’s payoff strictly increases from α ( w h − w e ) + ( 1 − α ) π h ( n ) for some α = 1 2 , 1 3 , … , 1 n − 1 to π i ( n ) , and for all other b ∈ B \ B ˆ , h’s payoff either increases or remains the same. Hence, h’s expected payoff from σ ′ h is greater than h’s expected payoff from σ h , contradicting σ being a Nash equilibrium.□

Proof of Lemma 3 Let any pair σ i , σ i ′ be such that Pr σ i [ b i = w e ] = Pr σ i ′ [ b i = w e ] for all bidders participating in preemption. For each S ⊆ H , let Pr [ S ] be the probability that all incumbents in S bid w e , and all the other incumbents in H \ S bid an amount less than w e . That is,

Part 1.

| H | = 2 . We prove the existence of a unique solution of (14) which gives ( p 1 * , p 2 * ) ≫ 0 . Note that (14) is composed of two equations below:

The unique solution is given by x 1 * = 2 ( λ 2 − 1 ) and x 2 * = 2 ( λ 1 − 1 ) . Since ( λ 1 , λ 2 ) ≫ ( 1 , 1 ) , ( x 1 * , x 2 * ) ≫ 0 , which provides the unique FPRP equilibrium.

Part 2.

| H | ≥ 3 . Let H ≡ { 1 , … , | H | } . Assume that all incumbents in H have identical entry-loss rate λ 0 . If there is an FPRP ( x h ) h ∈ H , then by Proposition 2 and (14), there is x 0 * such that x 0 * = x h for all h ∈ H and x 0 * is a solution of,

Note that Φ ( x 0 , … , x 0 ) − λ 0 takes a negative value at x 0 = 0 , and it is continuous, strictly increasing and unbounded above. Hence, there exists a unique x 0 * > 0 such that Φ ( x 0 * , … , x 0 * ) = λ 0 . Let x * ≡ ( x 0 * , … , x 0 * ) and λ * ≡ ( λ 0 , … , λ 0 ) . For each i ∈ H , each λ ≡ ( λ i ) i ∈ H , and x ≡ ( x i ) i ∈ H , let F i ( λ , x ) ≡ Φ ( x − i ) − λ i and F ( λ , x ) ≡ ( F i ( λ , x ) ) i ∈ H . Note that

and for all i , j ∈ H , ∂ Φ ( x − i * ) / ∂ x i = ∂ Φ ( x − j * ) / ∂ x j = c . Then

Then using (25) and applying Implicit Function Theorem for F ( λ , x ) at ( λ * , x * ) , there is an open set U ⊆ ℝ + | H | containing λ * ∈ U and a unique continuously differentiable function g : U → ℝ + | H | such that for all λ ∈ U , F ( λ , g ( λ ) ) = 0 . □

Proof of Proposition 4

Part 1.

Assume | S | = 2 and max i ∈ H λ i ∈ S . Let h ∈ arg max i ∈ H λ i and S ≡ { h , g } . From the proof of Proposition 3, there exist strictly mixed strategies ( x h * , x g * ) of two incumbents in S such that

For all i ∈ H \ S , let x i * ≡ 0 . It is optimal for i to choose a non-deterring bid with probability 1 ( x i * = 0 ) if (13) holds. Note that

where the second equality follows from (26). Since λ h − λ i ≥ 0 , we have Φ ( x − i * ) − λ i > 0 . Hence (13) holds. The uniqueness part follows from the uniqueness of the solution for (26).

Part 2.

Let S ⊆ H . Suppose that all incumbents in S have an identical entry-loss rate λ 0 . Consider λ * such that for all i ∈ S , λ i * = λ 0 and for all j ∈ H \ S , λ j * ≤ λ 0 .

A profile of binary preemptive mixed strategies p * is a symmetric PPRP equilibrium with the set of participating incumbents S if (i) for all i ∈ H \ S , p i * = 0 , (ii) there is p 0 * > 0 such that for all i ∈ S , p i * = p 0 * and x i * ≡ p 0 * / ( 1 − p 0 * ) and (iii) p * satisfies both (13) and (14).

Note that when all incumbents in S choose identical p 0 and x 0 = p 0 / ( 1 − p 0 ) , (14) can be rewritten as follows:

Existence of x 0 * satisfying (27) is straightforward. Let p 0 * be given by x 0 * = p 0 * / ( 1 − p 0 * ) . For all i ∈ S , let x h * ≡ x 0 * and p i * ≡ p 0 * and for all i ∈ H \ S , let x i * ≡ 0 and p i * ≡ 0 . Then p * satisfies (14). We show that p * is a symmetric PPRP equilibrium with the group of participating incumbents S. We have only to show that p * satisfies (13), which establishes that it is optimal for the incumbents in H \ S to choose non-deterring bids. Let i ∈ H \ S . Since x i * = 0 ,

where the third equality follows from (27). Since the difference inside of the bracket is positive and and by assumption λ 0 − λ i * ≥ 0 , we have Φ ( x − i * ) − λ i * > 0 , which shows (13).

Proof of Corollary 2

By a binomial expansion, (17) can be rewritten as

where x is a function of | S | .^[8] On the other hand, the entry probability of the PPRP is written as

To consider changes in the entry probability with respect to changes in | S | explicitly, denote x ( | S | ) . The derivative of ( 1 + x ) | S | − 1 with respect to | S | then yields

where d x ( | S | ) d | S | can be derived implicitly from (28). We obtain the negative sign since - ln 1 + x | S | ( 1 + x | S | + x | S | is equal to zero for x ( | S | ) = 0 , and strictly decreasing for all x ( | S | ) ≥ 0 .□

Proof of Proposition 5 First, examine case (i): λ f > λ l . By the monotonicity result of Proposition 2, x l * > x f * , and (18) implies that x f * > 0 , so x l * > x f * > 0 . It remains to show that the other incumbents optimally choose a non-deterring bid with probability 1. The payoff of each j ∈ H \ S

where the the equality follows from (18). Since the first term must be positive, and λ l − λ j > 0 , we have Φ ( x − j * ) − λ j > 0 , implying that each incumbent j ∈ H \ S chooses the deterring bid with zero probability.

However, for case (ii), λ l > λ f , the monotonicity result of Proposition 2 implies x f * > x l * , and furthermore, for a sufficiently great λ l , we have a corresponding high value of x f * from (18), which can produce a negative value of x l * from (19). Hence, in this case, the existence of a PPRP equilibrium is not guaranteed.□

Proof of Proposition 6

Part 1.

By integrating both sides of a binomial expansion ∑ k = 0 | S | - 1 | S | - 1 k z k = 1 + z | S | - 1 , we obtain

Hence we obtain (22) from (18). Similarly, by integrating both sides of a modification of a binomial expansion ∑ k = 0 | S | - 2 | S | - 2 k z k + 1 = z 1 + z | S | - 2 , we have

The first term in the left-hand side of (19) can be rewritten as

Hence, applying the previous binomial expansion, we get

which leads to

The second term of the left-hand side of (19) can be rewritten as in the left-hand side of (22). Together, we get (23).

Now, using (22), (23) can be written as follow:

which leads to

The difference in the entry probabilities of the symmetric equilibrium excluding the leader and the leader-follower equilibrium, namely the difference between (21) and (24) is given by

Therefore, λ l   x l * + λ f ( | S | − 1 ) ( x f * − x ˆ f ) < 0 (or > 0 ) implies that exclusion of the leader increases (or decreases, respectively) the entry probability.

Part 2.

If a leader-follower PPRP equilibrium with the set S does not exists, there exists a leader-follower PPRP equilibrium with a subset of S. At least one PPRP with the leader exists with one follower by Proposition 4 (i). Fix λ f . Then, for any PPRP with a subset S ˆ of S, there exists a corresponding λ l value such that the entry probability from limiting the leader is higher than the PPRP if the leader’s probability x l * is equal to zero. In this case, from (19) or (23), each follower’s x f * is identical to that of a symmetric equilibrium case with S ˆ . From Corollary 2, we know that the entry probability from limiting the leader is higher than this PPRP with x l * = 0 .