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Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Revealing Private Information in a Patent Race

Pavel Kocourek1 February 15, 2020

1pk1050@nyu.edu

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Consider competition of Apple and Samsung in patenting a new smartphone technology. Samsung makes an intermediate breakthrough on the way to a patent. Should Samsung disclose the breakthrough?

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Patent Race

Two firms compete to patent a specific product or technology. The breakthroughs arrive in a random fashion; the first firm to make two breakthroughs wins the patent, the other firm loses. Most of the patent race literature makes the questionable assumption that firms observe each others progress in the

• race. Should the firms disclose their breakthroughs? If not,

what is the dynamics of the patent race then?

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Assumptions

Two consecutive breakthroughs needed to patent. Patenting is public, the game ends. Each firm continuously chooses its effort that is equal to its hazard rate of making a breakthrough. The research happens in secrecy. (Effort and breakthroughs are not observable.) Each firm has the option to disclose its first breakthrough (being successful). Verifiable. No technological spillovers.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Intuition

Under complete information: If Apple is unsuccessful, Samsung’s success discourages Apple’s effort. If Apple is successful, Samsung’s success encourages Apple to hurry up. Under private information: Samsung does not observe Apple’s success. Samsung gets increasingly confident that Apple is successful over time. Samsung has decreasing incentive to disclose its success

• ver time.

After observing Apple’s disclosure, Samsung does not want to disclose its breakthrough.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Results

Without revelation: each firm drops its effort until its first breakthrough, after which its effort jumps up and keeps increasing. A firm never discloses after observing disclosure of its rival. Unique symmetric Nash equilibrium. The type of equilibrium depends on research difficulty. If research is difficult, then the first firm to make a breakthrough reveals it instantly; easy, then firms never reveal; moderate, then player’s reveal with a mixed strategy. Voluntary revelations is better for welfare than no revelation or mandatory revelation.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Literature

Complete information version of a patent race: Harris and Vickers (1987), Grossman and Shapiro (1987). Secrecy versus Patenting: Levin et al. (1987), Kultti et al. (2007) Two players, hidden effort choice, technological uncertainty: Bonatti and Horner (2011), “Collaborating”. Closest Related Study: The job market paper of Gordon (2003), “Publishing to Deter in R&D Competition” (Unpublished).

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Outline

1

Patent Race Under Secrecy

2

One Player Known to be Successful

3

Revealing Breakthroughs

4

Welfare and Policy

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Table of Contents

1

Patent Race Under Secrecy

2

One Player Known to be Successful

3

Revealing Breakthroughs

4

Welfare and Policy

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Patent Race Under Secrecy

Firm’s state is its private information. When a firm patents, it is a common knowledge. Effort is not observed. No option to reveal.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

The Race

Infinite horizon continuous time game with two players. Each player perpetually chooses any positive level of research effort. To win the patent, a firm has to be the first to make two consecutive discoveries. A firm is in state 0: at time t = 0; state 1: after making the first breakthrough; state 2: winning the patent – the game ends.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Breakthroughs and Patenting

Denote xj

t ∈ {0, 1, 2} and ej t ∈ [0, ∞) player j’s (j ∈ {A, B})

state and effort at time t ≥ 0, respectively. Initially, xj

0 = 0. At any time, Player j’s effort is the hazard

rate of making a breakthrough: P[xj

t+∆t = xj t + 1] = et∆t + o(∆t).

Player j patents when he makes the second discovery, let τ j be his patenting time: τ j = inf{t ≥ 0 : xj

t = 2}.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Expected Payoff

Player j incurs flow cost that is quadratic in effort, c(e) = 1

2αe2,

α > 0; and if he wins, he receives the price v > 0 of the patent; future payoffs are discounted at rate r > 0. His expected payoff is EUj = E τ − exp(−rt) · c(ej

t)dt

• R&D expenses

+ exp(−rτ) · 1j wins · v

• value of patent

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Normalization

Parameters: v . . . value of the patent; α . . . effort cost multiplier; r . . . discount rate. Choosing appropriate units of value and time, we can achieve v ′ = 1, α′ = 1. Then r ′ = αr v represents the research difficulty.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Continuation Value

Player j’s current continuation value depends only on his state x ∈ {0, 1} and time t ≥ 0, − ˙ v x,j

t

= max

e≥0

• (v x+1,j − v x,j

t )e − 1 2(e)2 − (r + ψ−j)v x,j t

• where ψ−j

t

is the hazard rate with which the rival patents at t. FOC implies ex,j

t

= v x+1,j

t

− v x,j

t . Then

− ˙ v x,j

t

= 1

2

• v x+1,j

t

− v x,j

t

2 −

• r + ψ−j

t

• v x,j

t .

Solution Concept: Nash Equilibrium (NE ≡ PBE ≡ MPE)

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Posterior Belief

Player −j has posterior belief about his rival being successful, pj

t = P[xj t = 1| none patented yet].

By Bayes rule, ˙ pj

t = (1 − pj t)(e0,j t

− pj

te1,j t ).

Rival patents with hazard rate ψj

t = pj te1,j t .

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

ODE

Proposition Any Nash equilibrium is characterized by the ODE − ˙ v 1,j

t

=

1 2

• 1 − v 1,j

t

2 −

• r + p−j

t e1,−j t

• v 1,j

t

− ˙ v 0,j

t

=

1 2

• v 1,j

t

− v 0,j

t

2 −

• r + p−j

t e1,−j t

• v 0,j

t

˙ pj

t

=

• 1 − pj

t

• e0,j

t

− pj

te1,j t

• ,

with e1,j

t

= 1 − v 1,j

t

and e0,j

t

= v 1,j

t

− v 0,j

t , and the initial

condition pj

0 = 0, and the restrictions 0 ≤ v 0,j t

≤ v 1,j

t

≤ 1 and pj

t ∈ [0, 1], for all t ≥ 0, j ∈ {A, B}.

Not an initial value problem.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Solution Method

The challenge: initial conditions v 1

0 and v 0 0 unknown;

an initial error grows exponentially over time. Going back in time: look for solutions that converge to a critical point; go back in time from the critical point,. . . but how? Getting out of the critical point: jump out so that the solution will converge back; go along the eigenvector related to the negative eigenvalue of the Jacobean (at the critical point).

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Unique Symmetric Nash Equilibrium

Theorem The patent race with private information has unique symmetric Nash equilibrium. Uniqueness any solution converges to a critical point; the critical point is unique; the Jacobian at the critical point has unique eigenvalue with negative real part. Existence: Going back in time the inequalities 0 ≤ v 0,j

t

≤ v 1,j

t

≤ 1 are preserved; pt eventually reaches 0.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Posterior Belief

Proposition The posterior belief pt about rival being successful steadily grows over time up to its steady-state value p∗ < 1. Why? The dynamics of the posterior is ˙ pt =

• 1 − pt
• e0

t − pte1 t

• ,

where e0

t and e1 t converge to steady-state values e0 ∗ and e1 ∗,

respectively. Since e0

∗ < e1 ∗, pt converges to p∗ = e0 ∗/e1 ∗ < 1

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Numerical Solution for r = 0.1

t

2 4 0.2 0.4 0.6 0.8 1

Posterior Probability

p

t

2 4 0.2 0.4 0.6 0.8 1

Value Functions

v1 v0

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Efforts

Proposition A firm decreases its effort over time until it makes the first discovery; then its effort jumps up and keeps increasing.

t

0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1

Sample Effort Trajectory

et

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Observations

Values and efforts as functions of posterior belief pt. (Dotted lines represent corresponding variables that a monopolist would choose.)

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Table of Contents

1

Patent Race Under Secrecy

2

One Player Known to be Successful

3

Revealing Breakthroughs

4

Welfare and Policy

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Asymmetric Private Information

Player A known to be only one step away from patenting (in state 1). Player B’s state unknown. With probability ˆ p in state 1 at t = 0.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Asymmetric Private Information

Proposition Suppose that player A has disclosed being in state 1. Any equilibrium is then given by the ODE − ˙ v 1A =

1 2(1 − v 1A)2 − (r + pBe1B)v 1A

− ˙ v 1B =

1 2(1 − v 1B)2 − (r + e1A)v 1B

− ˙ v 0B =

1 2(v 1B − v 0B)2 − (r + e1A)v 0B

˙ pB = (1 − pB)(e0B − pBe1B), with e1A = 1 − v 1A, e1B = 1 − v 1B, e0B = v 1B − v 0B, and the initial condition pB

0 = ˆ

p and restrictions 0 ≤ v 1A

t

≤ 1 and 0 ≤ v 0B

t

≤ v 1B

t

≤ 1, for all t ≥ 0.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Observations

Suppose that player A is known to be in state 1. Unique Nash Equilibrium, as in the symmetric case: Posterior belief about player B’s success raises monotonically over time. Player B drops effort until making the first breakthrough, after which his effort jumps up and keeps increasing. Thanks to the asymmetry: v 1B

t

< v 1A

t

. . . the informed player is less optimistic; e1B

t

> e1A

t

. . . the informed player exerts higher effort.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Observations

The informed player is less optimistic, so that he exerts higher effort. pB

0.5 1 0.3 0.35 0.4 0.45 0.5 0.55

Continuation Values

v1A v1B

pB

0.5 1 0.45 0.5 0.55 0.6 0.65 0.7

Efforts

e1A e1B

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Who is Better off?

Define v 1A,(11) the continuation value of player A from the perspective of an outside observer who knows that player B is in state 1. Lemma Conditioned on both players being in state 1, the informed player is better off: v 1B

t

> v 1A,(11)

t

. Recall that v 1B

t

< v 1A

t

implies that e1B

t

> e1A

t .

The informed player is facing a more optimistic, hence less aggressive, rival. In addition, he can use the information to his advantage, so he is better off.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Table of Contents

1

Patent Race Under Secrecy

2

One Player Known to be Successful

3

Revealing Breakthroughs

4

Welfare and Policy

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Patent Race with Revealing

In addition to having private information about its state a firm has the option to disclose its breakthrough (being in state 1). Revelation is truthful, and yet does not leak any information about how the discovery was made – no imitation concerns.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Symmetric Sequential Equilibrium

Sequential equilibrium can be found by backward induction:

1

solve the sub-game after both players have revealed;

2

solve the sub-game after exactly one of the players have revealed;

3

solve the game before anyone have revealed.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Never Reveal Second

Proposition A player never reveals after the rival has done so.

pB

0.2 0.4 0.6 0.8 1 0.3 0.31 0.32 0.33 0.34 0.35

Continuation value

v1B(pB)

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Never Reveal Second

Proposition A player never reveals after the rival has done so. Proof: Consider the subgame after A has revealed success in equilibrium with B not revealing, B has no incentive to reveal; in any equilibrium in which B had strategy to reveal with a positive probability, B would benefit from postponing the revelation by ∆t.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Before Anyone Has Revealed

Suppose none of the players have revealed yet and player A has made a breakthrough – revealing it has the following impacts:

1

it discourages player B in case that he is unsuccessful;

2

it encourages player B in case that he is successful;

3

it prevents player B from revealing.

1 2 3 4 5 0.5 1

Efforts before and after revealing

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Unique Symmetric Sequential Equilibrium

Never Reveal Reveal with a delay Reveal Instantly

αr v

0.1113 0.1705 A does not reveal as he expects B to succeed soon. A delays revealing to wait for B to reveal. A reveals its breakthrough to discourage B who is likely

• unsuccessful. (Then pt = 0.)

Player A’s strategy (assuming B has not revealed):

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

No Revelation

Result (partially numerical) For r small, there is a symmetric sequential equilibrium in which no player ever reveals. Condition: If both are expected never to reveal, none is tempted to reveal.

0.2 0.4 0.6 0.8 1 0.3 0.4 0.5 0.6

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Instant Revelation

Result (partially numerical) For r large, there is a symmetric sequential equilibrium in which the first successful player reveals instantly. Condition: Given that both players are expected to reveal instantly, none is tempted to avoid revealing. Note that a player is certain that the rival is unsuccessful unless he has revealed.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Overlap?

Can the two types of equilibria, no revealing and instant revealing, coexist for the same r?

• No. To the contrary, there is a gap between the two.

Expectation of rival’s revelation discourages player’s revelation.

r

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 #10-3

• 15
• 10
• 5

5

Incentives to reveal assuming:

No Revealing Instant Revealing

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Mixed Revealing

Result (partially numerical) For r moderate there is a sequential equilibrium in which players mixing over revealing and eventually stop revealing at all. Condition: It is possible to mix over revealing to make the rival indifferent.

0.02 0.04 0.06 0.482 0.484 0.486 0.488 0.49 0.492 Payoffs if rival didn't reveal

Revealed Not Reveal

0.5 1 1.5 0.25 0.3

Hazart-rate of revealing

0.5 1 1.5 0.1 0.2 0.3

Posterior

With revealing Without revealing

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Variable Difficulty of the Two Stages

Let α0 and α1 be effort cost multipliers in the two stages.

0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4

Revealing

Immediate Delayed Never

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Variable Difficulty of the Two Stages

Higher difficulty of either stage increases the incentive of a player to reveal success. Higher difficulty of the first stage implies: It takes the rival longer to succeed and so he gets discouraged for a longer time. Higher difficulty of the second stage implies: The discouragement of rival when being unsuccessful is “stronger” than the encouragement of the rival when being successful.

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Table of Contents

1

Patent Race Under Secrecy

2

One Player Known to be Successful

3

Revealing Breakthroughs

4

Welfare and Policy

Patent Race Under Secrecy One Player Known to be Successful Revealing Breakthroughs Welfare and Policy

Welfare and Policy Implications

0.05 0.1 0.15 0.2 0.25 99.5 100 100.5 101

Total Payoff (% of public)

Secrecy First Reveals 0.05 0.1 0.15 0.2 0.25 99.8 100 100.2 100.4 100.6

% Value of Patent (% of public)

Secrecy First Reveals

Summary

In a patent race with private information (and no revelation): A player gets increasingly discouraged from R&D effort by expecting rival to be ahead, but once he makes the first breakthrough his effort jumps up and keeps increasing. If firms have the option to reveal breakthroughs: low difficulty . . . no firm ever reveals; high difficulty . . . the first innovator reveals instantly; medium difficulty . . . mixed strategy / delay. In addition to concerns about information spillovers, firms are discouraged from patenting small inventions as it might hurry up the rival.

Extensions

More than two players – first player reveals instantly. Three stages – does a player want to reveal a minor success,

• r wait to reveal the major one?

Other factors: Encouraging to reveal: spontaneous revealing, idiosyncratic technological uncertainty,. . . Discouraging from revealing: technological spillover, shared technological uncertainty, . . .