Meaning step one. The latest harmony within our design try a Markov Finest Harmony for example one to, at every months t , the proper RA always.
I select a great Markov Best Equilibrium in the same way one to the latest balance try “memoryless,” which is, the strategy of one’s proper RA merely hinges on the modern history of their challenger and itself. This new harmony is also “symmetrical,” due to the fact method function of both RAs (if they’re both proper) is similar. not, the fresh new RAs do not need methods at exactly the same time.
Let RA1 be a strategic RA and let Vt(q1, q2) 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 Vt(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.
In order to obtain an analytical substitute for this video game, we make a good simplifying expectation one to p
Proposition 1. There exists a unique x1 , where 0 ? x1 ? 1 , given that Vt(q1, q2) is an increasing function in q1 .
Intuitively, it is easy to see from Equation (8) that Vt(q1, q2) is linear in x1 . This ensures that RA1’s maximization problem has a unique solution.
Proposal 2 means that a does telegraph dating work strategic RA always provides GR so you can a great venture. For the reason that it becomes a lower life expectancy shell out-out of whether or not it deviates out of this means and supply an effective NR so you’re able to a investment. The fresh suggestion follows directly from the fresh spend-out-of design of your RAs as well as the viewpoints.
Corollary 1. Assume pG < 1 . Then the equilibrium strategy of the strategic RA is always positive, that is, it inflates ratings with positive probability.
Corollary dos. Guess the fresh new design ends in several months T. Then the balance strategy of your own strategic RA are x = 1 at t = T ? step 1, T .
We currently present a logical provider inside the a finite months means. I solve the brand new model numerically when you look at the infinite vista inside Part 5.
4 Finite Horizon Provider
I suppose the latest model can last for about three periods, t = step one,dos,3 , plus the RAs optimize the requested overall income over the about three periods. I compute the newest equilibrium approach of the RAs playing with backward induction. I know that proper RA are often lie in the past two episodes, because the found into 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 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.