Definition step step one. The fresh equilibrium within our model try a Markov Best Harmony including you to, at every period t , the new proper RA always.
I select an effective Markov Perfect Equilibrium in the sense one to brand new harmony was “memoryless,” that is, the strategy of your proper RA just relies on the present day history of its enemy and you can by itself. The newest harmony is even “symmetrical,” since the approach reason for each other RAs (when they both strategic) is the same. However, the latest RAs don’t bring steps while doing so.
Let RA1 be a strategic RA and let Vt(q1, qdos) 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.
To help you derive an analytical option to this video game, we make a great simplifying expectation you 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.
Suggestion 2 implies that a strategic RA constantly gets GR in order to a beneficial endeavor. Simply because it will become a diminished pay-out of whether or not it deviates from this means and offer good NR in order to a investment. The offer pursue directly from the brand new pay-off framework of your RAs in addition to philosophy.
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 2. Suppose the model ends in period T. Then balance approach of your own proper RA try x = 1 on t = T ? step 1, T .
We now expose a logical service in https://www.datingranking.net/spicymatch-review/ a limited several months setting. We resolve new design numerically during the infinite vista from inside the Point 5.
4 Finite Vista Solution
We guess the model can last for about three attacks, t = step one,dos,step three , while the RAs maximize its expected complete money along side about three attacks. I calculate brand new equilibrium method of your RAs having fun with backward induction. We already know just that proper RA will always lay inside the the past a few periods, due to the fact found when you look at 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.