Robust Policy Schemes for Differential R&D Games with Asymmetric Information

Abstract

I consider a general differential R&D game with finite set of firms which may generate a multiplicity of strategically driven outcomes like prevention of entry or strategic delay. It is assumed that while interacting firms are fully aware of their potentials, the government is uncertain over the true state of the industry and thus may be unable to predict such strategic behavior. The choice function over the set of potential outcomes is defined, and robust welfare optimal and individually optimal regimes are compared. The belief of the planner over the market shapes the set of potential policy schemes. The ordering of such schemes with respect to uncertainty, costs and welfare-improving potential is established. At last, the optimal level of robustness for a given policy is found.

Appendices

A Proof of Proposition 1

Proof

The proof simply follows from the definition of value functions $\Pi ^{m}_{j}$ and the fact that the choice function and the selector exist for any finite (and countable) collection of arbitrary sets satisfying the axiom of choice.

In particular, $\Pi ^{m}_{j}$ are functions of initial state of the game $\delta (0)$ only and thus their differences too. Each of the intervals $[\delta ^{j}_{z}(m);\delta ^{j}_{z+1}(m)]$ denotes the range of initial value $\delta (0)$ for which the difference in values of the game for the leader j across a given outcome i and some other outcome m has constant sign (positive or negative), since $\delta ^{j}_{z}(m);\delta ^{j}_{z+1}(m)$ denote the z-th and $z+1$th roots (zeros) of this difference for j as an algebraic expression. By Assumption 2, there are at most countably many such roots for each value function and hence at most countably many roots for differences of these values. Consider next only those such intervals that imply $\Pi ^{i}_{j}(\delta (0))-\Pi ^{m}_{j}(\delta (0))\ge 0$. If $\delta (0)$ belongs to one of such intervals, it implies that outcome i is preferred to an outcome m.

We next compare i to all other m outcomes and consider those intervals where profit difference is positive. The intersection of all such intervals would give a range of $\delta (0)$ where outcome i is preferred to all other feasible outcomes. At last, we need to consider a union of all these intersections since there might be multiple of those.

The choice function $\Theta ({\mathcal {F}})$ selects the outcomei among all feasible ones if and only if the given initial state $\delta (0)$ belongs to the set of values where the value for j under outcome i is greater than under any other outcome, as denoted by the argmax term. At this point, we need axiom of choice which guarantees that it is always possible to select an element within a set given some criterion (the selector function), whereas the selector is given by the above procedure on intervals between zeros of value functions.

Example: Consider as an illustration the example with only three possible outcomes, a, b, c, and assume for simplicity all values are given as polynomials of 3d degree in $\delta (0)$. In this case, for an outcome a to be optimal, it is needed that values differences $\Pi ^{a}-\Pi ^{b}, \Pi ^{a}-\Pi ^{c}$ remain positive at the given $\delta (0)$ level. So consider intervals between roots of those polynomials (four intervals for the case of real roots) for each of alternatives. Assume $\delta (0)\in \{[\delta _{1}(b),\delta _{2}(b)|\Pi ^{a}-\Pi ^{b}\ge 0\}$ meaning that a is preferred to b once $\delta (0)$ is in this range. Then, find the same interval for c alternative. If the intersection of those two ranges is non-empty and the union of such intersections contains $\delta (0)$, outcome a is optimal in comparison with the other two, that is, the selector function gives a as an argmax of value function w.r.t. the set of outcomes (and not $\delta (0)$ value itself). $\square$

B Proof of Proposition 3

Proof

• 1. The condition $\Theta _{\epsilon >\epsilon ^{R}_{k}(i,s)}({\mathcal {F}}_{\Sigma ^{\epsilon }_{k}})\ne \Theta _{\epsilon <\epsilon ^{R}_{k}(i,s)}({\mathcal {F}}_{\Sigma ^{\epsilon }_{k}})=s$ means that at the noise level $\epsilon ^{R}_{k}(i,s)$ the outcome of the game believed to be realized is changing, so this is a threshold distinguishing the level under which subsidy from i to s is robust from not being robust. Thus, by definition the given policy scheme is robust up to this level of noise. If such a change in believed outcome does not exist up to the maximal noise level $\epsilon ^{W}_{s}$ associated with the outcome s (and chosen by the planner), the given scheme will always have an effect in switching the outcome from i to s.

• 2. The same claim follows for welfare functions: Once under a given noise level $\epsilon ^{S}_{k}(i,s)$ the preferred outcome changes (the choice function $\Psi$ of the planner changes its value), this defines a threshold up to which the given policy is improving the (expected) welfare. Again, if such a change in choice function is not observed for any noise level up to the maximal one, given subsidy scheme is always welfare improving.

• 3. Combining two previous arguments, we observe that the given policy scheme will be implementable (that is, both robust and welfare improving) only for noise level being under both above-defined thresholds (if they exist).

• 4. Now we form a set of all admissible policy schemes as defined by the previous point for a given noise level and two outcomes i, s. If this set is non-empty, then there is at least one policy scheme for a noise level up to $\epsilon ^{W}_{s}$ which is both robust and welfare improving in switching the regime from i to s. Then, there is a problem of selecting the best of such admissible schemes (since they are all only welfare improving but not optimal). We then may define the welfare associated with each of such policy schemes and formulate choice criteria in the same way as in previous propositions of the paper: the policy scheme which yields the best worst case welfare under given noise level $\epsilon$ is selected as the most appropriate one; see (34), which defines the selection criteria in the set of these admissible policy schemes.

• 5. Fix the noise level for which a given policy scheme is optimal. Once we allow this value to vary, it is straightforward that the choice function (34) will change its value (i.e., will select a different scheme) at some level of noise. This level is the next threshold, where the scheme $\Sigma ^{\epsilon }_{x}$ ceases either being robust or welfare improving and the set of admissible schemes shrinks. The fact that the set will shrink under increasing $\epsilon$ is implied by the min operator over $\epsilon$ in (34) formulation. Thus, there is an increasing sequence of $\epsilon ^{*}$ such that the previously admissible scheme stops to be such and the $\Lambda$ selector will yield a different result.

• 6. By selecting at each such a threshold a new policy scheme the sequence $\Sigma ^{*}$ is formed. Since we have countably many schemes, the increasing sequence can be constructed out of it. It follows that each next element is better then previous ones under given noise level, since otherwise those previously selected elements will remain optimal.$\square$

C Proof of Corollary 4

Proof

• 1. Once the belief of the social planner is such that only delay may realize, this outcome may be switched to the simultaneous one in a variety of ways. The set $\Sigma _{\epsilon }(d,*)$ defined as in Proposition 3 excludes schemes $\Sigma _{+}, \Sigma _{o}$ since even in the extreme case of no information except the expected realization the scheme $\Sigma _{-}$ is robust and by (52) it is less costly than the other two constant schemes. Hence, $\Sigma _{\epsilon }(d,*)=\{\{\Sigma ^{\epsilon }_{\mathrm{min}},\Sigma ^{\epsilon }_{-},\Sigma ^{\epsilon }_\mathrm{suff},\Sigma ^{\epsilon }_{\mathrm{max}}\}\}$. Then, we start with robustness level $\epsilon <\min \{\epsilon ^{R}_{\mathrm{min}},\epsilon ^{S}_{\mathrm{min}}\}$, for which, by Proposition 3 all the set is admissible. Then, the most welfare-improving scheme is selected which is by (52) scheme $\Sigma ^{\epsilon }_{\mathrm{min}}$, giving case a. Once we increase the confidence interval to the level $\min \{\epsilon ^{R}_\mathrm{suff},\epsilon ^{S}_\mathrm{suff}\}$, this scheme stops being robust and the set $\Sigma _{\epsilon }(d,*)$ looses one element. Thus, the next scheme is selected, which is $\Sigma ^{\epsilon }_\mathrm{suff}$. The process continues until the maximal level of noise is reached, under which only the constant subsidy remains robust.

• 2. Once the belief is different and permits multiple other outcomes except the delay, minimal subsidy cannot be robust since it does not prevent firm A from strategic pricing and the same is true for its constant counterpart. At the same time, since (52) the scheme $\Sigma _{+}$ is clearly dominated by the $\Sigma _{o}$ and thus is also excluded from the admissible set. We get $\Sigma _{\epsilon }(m,*)=\{\Sigma ^{\epsilon }_\mathrm{suff},\Sigma ^{\epsilon }_{o},\Sigma ^{\epsilon }_{\mathrm{max}}\}$. We next again apply Proposition 3 increasing the uncertainty tolerance level $\epsilon$ and selecting at each of the threshold the next optimal scheme after the shrinkage of the admissible set.

• 3. At last, if there is not enough information for the planner to believe in any particular outcome of the game, only the constant maximal scheme is robust and can prevent strategic pricing. This would be the case if the robustness level is above the one enabling to select the belief in any particular outcome, $\max \{\epsilon ^{O}_{m\in {\mathcal {F}}}\}$ but still under the threshold where the planner believes in the social optimality of simultaneous development, $\epsilon ^{W}_{*}$. The policy scheme is welfare improving (and thus optimal since the set of admissible subsidies is a singleton) if additionally it is robust; i.e., the noise level does not exceed its robustness level.$\square$