1.2 MARCO CONCEPTUAL
1.2.1 REGLAMENTO TECNICO ECUATORIANO RTE INEN 013:2013 (PRIMERA REVISION)
I now return to the original model where A can build a reputation for being of
commitment type. Define ψ(pA, δ) =
(1−δ)(1−pA) pA
Theorem 6. Given parameters (pA, δ) ∈ (0,1)2 satisfying ψ(pA, δ) ≤ 1+δδ4 there
exists a PBE which gives payoffs UA= 1−ψ and UB=ψ.
Noticing that for any pA>0, limδ→1ψ(pA, δ) = 0 gives the following:
Theorem 7. For any ε > 0 and pA > 0, there exists δ¯ ∈ (0,1) such that for any
discount factor δ >δ¯, there is a PBE with payoffs UA>1−ε and UB < ε.
This means that, no matter how small the initial probability that A is the com-
mitment type, for high enough discount factors, A takes almost the whole pie.
The equilibrium constructed here has immediate agreement at period 0, and also
immediate agreement in any subgame starting fromAmaking an offer or threat. That
is, in even periods A makes an offer which B accepts, while in odd periods A makes
a threat which B accommodates. The offers in even periods satisfy the no slackness
condition; while there is a bit of slackness in odd periods but this is disappearing in the limit as δ→1.
Finding the no slackness equilibrium for the complete range of parameter values is
not practical5. However, logically we can make some conclusions about it. Slackness reduces the payoffs of player A and increases the payoffs of player B, since for every
PBE with slackness, there exists another PBE in whichAdemands slightly more and
B accommodates the demand. Therefore the conclusion of Theorem 7 must also hold
for the no slackness equilibrium.
The construction of the PBE of Theorem 6 relies upon the following property
in odd periods: if B was to deviate away from accommodating the offer, then his
next best action involves making an offer which A always rejects for sure would be
better for him than making an offer A might accept. This property makes it far
easier to calculate continuation payoffs off the equilibrium path, and hence avoids the difficulties encountered when trying to calculate the no slackness equilibirium. This property is shared by the no slackness PBE in the finite horizon version of this game and consequently much of the analysis is shared with this case.
Consider the finite horizon (N + 1 period) version: If by the end of period N
agreement still has not been reached then the game ends with both players receiving payoff of zero6, otherwise the model is the same as laid out at the start of Section 2. Lemma 8. Consider theN+1period finite horizon version with parameters(pA, δ, N)
satisfying 1−δNδ ≥ 1
pA. The PBE satisfying the no slackness condition has payoffs
UA= 1−δN(1−pA), UB =δN(1−pA)
The proof is by backwards induction argument and is relegated to the appendix,
but the idea is as follows: In even periods, there are no surprises as A makes the
highest offer which B should accept - that is the offer equal to the continuation
payoff if he rejects. The key to calculating this equilibrium is the analysis in odd
5There is a sequence 1> a
1(δ)> a2(δ)> a3(δ)> . . . > 0 such that the equilibrium strategies
constitute a different function of parameters for each interval (ai+1ai) which the belief may fall into
6Considering different disagreement payoffs does not change the structure of the equilibrium
periods. Consider what would have happened had A set the threat y = 1, so that
B must optimise by offering something AC rejects. The big question is then what
offers AR accepts. This depends on his continuation payoff which is an increasing
function of the beliefB attaches to Abeing of commitment type. If AR accepts with
probability 1 then after a rejection B believes A is of commitment type for sure,
meaning that A takes the whole pie in the continuation game. So any offer below δ
must be rejected by AR with positive probability, and lower offers are accepted with lower probability.
The proof shows that after an unreasonable threat likey= 1, given a high enough discount factor so that 1−δNδ ≥ 1
pA, player B does best by offering something thatA R
rejects for sure. This means that any threat made byAwhich leavesBwith more than his continuation payoff from causing a one period delay should be accommodated.
Therefore in equilibrium A sets a threat leavingB with pecisely this amount, and B
accommodates leading to immediate agreement.
The reason that for such a high discount factor B is better off offering something
A is certain to reject than offering something which elicits a mixed response from
AR is the following: If B offers something below the threat and AR accepts for sure
then after a rejection A is known to be the commitment type and so gains a payoff
of 1 in the continuation game. Thus any offer below δ must be rejected by AR with
some probability. If B offers something to which AR mixes between acceping and
rejecting, then the posterior beliefs of the commitment type increase, which lowers B’s continuation payoffs after a rejection. Given the paramter values assumed in Lemma 8, this reduction in continuation payoffs after a rejection outweighs the gain
from the possible immediate agreement. Player B would be better off δN (1−p
A)
deliberately delaying agreement until the last period, and then offering 0 which A
accepts with probability (1−pA). Lemma 8 says that for sufficiently high discount
power until the last period. This enables playerAto take almost the entire pie in the first period.
With Lemma 8 in place, the proof of Theorem 6 becomes a lot simpler. We can construst an equilibrium with payoffs as follows: in any odd period in which the belief of A being the commitment type is qo, the continuation payoff toB is δK−1(1−qo);
while in even periods with belief qe, the continuation payoff to B is δK(1−qe) for
some K ∈ R satisfying 1−δKδ ≥ 1
pA. That B cannot do better follows the same logic
as Lemma 8, while a form of threatening by beliefs is used to ensure that A cannot
profitably deviate - a deviation byAwould be interpreted as coming from the rational type.
Discussion of reputation effects
The result of Theorem 7, based on the PBE of Theorem 6 is that for any positive
chance of A being the commitment type, as players become increasingly patient, A
takes the whole pie, whereas with no probability of commitment type, the pie is
shared equally. Where does this dramatic shift come from which enables A to take
B’s half of the pie? As shown in Lemma 5, the direct effect of enabling A to become
committed to a threat with probability pA every odd period has an effect only in
proportion to the size of pA. So for small probabilities the effect is fairly negligible.
The explanation lies in the difference between the models studied in Lemma 5 and
Theorem 6 - that is the ability of A to build a reputation for being of commitment
type. WhenpAis small andδ →1 it is this reputation effect which accounts for almost
the entire half of the pie which would otherwise remain withB. Under the stochastic
commitment asumption if AR rejects a reasonable offer from B then this does not
gain him anything. In future periodsB still attaches the same probability toA being committed. However, in the main model, where players can be of commitment type, they can crucially build a reputation for being of commitment type. So rejecting a
seemingly reasonable offer can have the consequence that in future periodsB thinks
it is more likely that A will be of commitment type. This gives AR an incentive to
reject even very high offers with some probability, and lower offers with far greater
probability. SupposeA sets an unreasonable high threat and so B makes an offerAC
rejects, but thatARshould accept with mixed probability. Now there are two effects
working against B: firstly there is a significant probability that the offer is rejected, thus eroding his bargaining power, and secondly if rejection occurs there is a higher
subsequent belief that A is of commitment type and so the first problem intensifies
for future periods. Of course player A will not set such unreasonable threats in
equilibrium, but knowing the problemB faces, can exploit this by making the threat
as tough as B is willing to accommodate.