For notational convenience, i prevents the time subscript of them reputation-upgrading features

Definition step 1. New equilibrium in our design are a beneficial Markov Primary Harmony such as for example you to, at each several months t , new proper RA always.

We find a great Markov Finest Balance in the sense you to definitely brand new balance is “memoryless,” which is, the strategy of proper RA simply depends on the modern reputation of its opponent and you can in itself. The brand new balance is even “shaped,” as the strategy intent behind each other RAs (when they one another strategic) is the identical. Yet not, the brand new RAs don’t capture measures as well.

Let RA1 be a strategic RA and let V_t(q₁, q_dos) denote its discounted future profits, given its reputation q₁ and its competitor’s reputation q₂ , and let ? be the discount rate. The RA’s new reputation after it gives NR and the failure of a project following a GR are denoted by and , respectively. A successful project with a GR leaves the RA’s reputation unchanged. Note that and are functions of the strategy of the RA and its current reputation level.

The objective function of RA1 is to maximize V_t(q₁, q₂) , the strategy being x₁ . Note that, RA1’s strategy is only effectual when it rates a bad project. In all other cases, RA1’s strategy is inconsequential.

To help you obtain a logical solution to the game, we make a great simplifying expectation you to p

Proposition 1. There exists a unique x₁ , where 0 ? x₁ ? 1 , given that V_t(q₁, q₂) is an increasing function in q₁ .

Intuitively, it is easy to see from Equation (8) that V_t(q₁, q₂) is linear in x₁ . This ensures that RA1’s maximization problem has a unique solution.

Proposition 2 ensures that a strategic RA usually brings GR to an effective venture. This is because it gets a lesser spend-from whether it deviates from this means and provide a NR so you’re able to an effective venture. Brand new suggestion comes after straight from the latest pay-from construction of the RAs plus the viewpoints.

Corollary 1. Assume p_G < 1 . Then the equilibrium strategy of the strategic RA is always positive, that is, it inflates ratings with positive probability.

Corollary dos. Guess the new design results in period T. Then the harmony strategy of strategic RA are x = step 1 on t = T ? step one, T .

We have now introduce a logical solution in a finite months setting. We resolve the latest design numerically inside the unlimited panorama into the Area 5.

cuatro Limited Views Solution

I suppose the new design lasts for around three episodes, t = step 1,2,3 , plus the RAs optimize its requested total earnings over the three episodes. I compute the fresh new balance method of RAs playing with backward induction. I already know your strategic RA are often sit for the the last a couple symptoms, since the revealed within the Corollary dos.

As described in Section 3, we look for an equilibrium of the game by examining the trade-off facing RA1, that is, the difference between expressions (9) and (10). If the pay-off from lying is greater then x₁ = 1 , and we have a pure-strategy equilibrium in which RA1 always lies; if the pay-off from not lying is greater then x₁ = 0 and we have a pure-strategy equilibrium in which RA1 never lies; otherwise, we have a mixed-strategy equilibrium in which RA1 is indifferent between lying and not lying, given some prior beliefs about its strategy, that is, 0 < x₁ < 1 .

_G = 1 and ? = 1 . This assumption implies that the reputation of the strategic RA goes to zero if it gives a GR to a bad project since now every good pure mobile project succeeds and every bad project fails. This simplifies expressions (9) and (10) and allows us to derive the equilibrium strategy of RA1. This assumption is relaxed in Section 5.