The monetary policy-making game involves three players: the government (G), the central bank bureaucrat (B), and the median legislator of the governing coalition (L). L represents the government’s backbench legislators and coalition partners. The government may propose to retain the status quo or initiate a policy change, P. The policy change, if enacted, produces a new policy outcome. The central bank bureaucrat also makes a policy recommendation, either endorsing or criticizing the potential policy change. After observing the government’s policy choice and the bureaucrat’s policy recommendation, the legislator decides whether to approve or veto the policy change. I evaluate two possible scenarios. In the first, the government lacks any ex-post authority over the bureaucrat. In the second, the government can impose a penalty on the central banker.
Before the players, G, B, and L, choose their strategies, Nature selects the value of a random variable, D, which determines the outcome produced by enacting P. Variable D takes on two values, 1 (with probability d) and 0 (with probability 1–d), where 0 < d < 1. If D = 1, the enactment of P yields outcome p1, whereas if D = 0, the enactment of P produces outcome p2.
G and B know the value of D, while L does not. That is, the government and the bureaucrat know the consequences of a change in monetary policy, P. The legislator, on the other hand, is unaware of the potential consequences of a change in monetary policy. L’s beliefs about the probability of Nature making different choices are as follows:
q1 = p(D = 1)
q2 = p(D = 0).
Prior to observing the messages sent by G and B, L’s beliefs are q1 = d and q2 = (1 – d).
A strategy for G specifies his policy choice, whether to adopt a new monetary policy, P, or retain the status quo, sq, as a function of his policy goals and his information about P. Formally, G’s strategy, denoted G(##), indicates whether he recommends P or sq given the value of D. The first number gives G’s action (M = recommend a policy change P; Q = retain the status quo) when P leads to p1 (i.e., when D = 1). The second number indicates G’s action when P leads to p2 (i.e., when D = 0).
B’s strategy, B(##), indicates her policy recommendation based on her policy goals and her information about the consequences of a policy change P. The first number gives B’s action (E = endorse a policy change P; C = criticize a policy change) when P leads to p1 (i.e., when D = 1). The second indicates B’s strategy when P leads to p2 (i.e., when D = 0).
L’s strategy, L(####), specifies his vote, up or down, on a policy change P, based on the messages sent by G and B. The first number indicates how L votes (u = approve policy change P; d = veto P and retain the status quo) given that both G and B evaluate a policy change favorably (i.e., when L sees the messages M and E). The second number indicates L’s vote when G approves a policy change (G sends message M), but B criticizes a potential policy change (B sends message C). The third number indicates how L votes given that G recommends retaining the status quo (G sends Q), but B endorses a policy change (B sends E). Finally, the fourth number indicates L’s vote when both G and B make a recommendation to retain the status quo (G sends Q, B sends C).
L’s payoff, E(uL), reflects the realized outcome and is denoted by uL(.). L’s policy preferences are p1 > sq > p2. If D = 1 (i.e., if P leads to p1), then L prefers to enact P. If D = 0 (i.e., if P leads to p2), then L prefers the status quo. Therefore, the three possible payoffs are ordered as follows: uL(p1) > uL(sq) > uL(p2).
The payoffs of G, E(uG), are a function of two elements: the realized outcome, denoted uG(.), and L’s vote on policy. The relative magnitudes of uG(p1), uG(sq), and uG(p2) vary according to G’s policy preferences:
Order 1: p1 > sq > p2
Order 3: sq > p2 > p1
Order 4: P1> p2 > sq
Order 5: p2 > p1 > sq
Order 6: p2 > sq > p1.
G also incurs a cost k if L votes against his policy choice. This part of G’s payoff reflects the negative consequences of a legislative veto. The magnitude of k is larger than any possible gain in utility from achieving his preferred policy outcome. That is, G prefers to sacrifice his preferred policy outcome to avoid a legislative veto.
B’s payoffs, E(uB), differ between two scenarios, depending on the government’s ex-post authority. In the first scenario, E(uB) is a function of the realized outcome, uB(.), and L’s vote on policy. The relative magnitudes of uB(p1), uB(sq), and uB(p2) vary according to B’s policy preferences. As with G, B has six possible preference orderings.
B also incurs a cost, c, if L votes against B’s policy recommendation. The magnitude of c is larger than any possible gain in utility from attaining her preferred policy outcome. That is, B prefers to avoid a legislative veto even if she could ensure her most preferred policy outcome.
In the second scenario, the government has ex-post authority over the bureaucrat. In this case, B’s policy payoffs are a function of three elements: the realized outcome, uB(.), L’s policy vote and G’s policy recommendation. The first two components of B’s policy payoffs, uB(.) and c, are the same as in the first scenario. In this scenario, however, B incurs a cost, t, if the substance of B’s action differs from G’s policy recommendation. That is, if G chooses to pursue a policy change P (and sends a message, M), but B criticizes the policy choice (sends message C), B incurs cost t. If G chooses to retain the status quo (sends message Q), but B endorses a policy change (sends message E), B also incurs cost t. This cost t reflects the cost of ex-post action by the government against the bureaucrat, including dismissal, budget cuts, and so on. As with c, the magnitude of t is larger than any possible gain in utility from attaining her preferred policy outcome.
The sequence of moves is as follows:
1. Nature chooses D (i.e., whether a change in policy, P, leads to p1 or p2) and informs G and B.
Fig. A 1. Game tree: Government has no ex-post authority
2. The government chooses G(.), whether to pursue a policy change (M) or retain the status quo (Q).
3. The bureaucrat chooses B(.), whether to endorse a policy change (E) or criticize a policy change (C).
4. After observing G’s action and B’s action, the legislator chooses L(.), whether to approve a policy change (u) or veto it (d).
5. The payoffs are distributed to the players.
Figures Al and A2 give the extensive forms of the policy-making game. Figure Al indicates the form when the government has no ex-post authority over the bureaucrat. Figure A2 shows the form when the government has ex-post authority over the bureaucrat. In both scenarios, the extensive form is assumed to be common knowledge.
Fig. A2. Game tree: Government has ex-post authority
The equilibrium concept is sequential equilibrium. A sequential equilibrium in these two games meets the following criteria:
1. G’s strategy maximizes E(uG) given B’s and L’s strategies and Nature’s choice.
2. B’s strategy maximizes E(uB) given G’s and L’s strategies and Nature’s choice.
3. L’s strategy maximizes E(uL) given G’s and B’s strategies, Nature’s choice, and L’s beliefs about P. L’s beliefs are calculated using Bayes’ Rule wherever possible.
Additionally, L’s off-equilibrium beliefs also satisfy the “intuitive criteria” (Banks and Sobel 1987; Cho and Kreps 1987). That is, L assigns zero probability to types of G and B who certainly lose by deviating from whatever equilibrium strategy they are supposed to be playing. In this game, L’s off-equilibrium beliefs differ in three cases, depending on the preference ordering of the senders. In the first two cases, off-equilibrium messages contain new information that may affect L’s choice of strategy:
1. If one of the senders has a preference ordering that is completely opposite L’s ordering, p2 > sq > p1, off-equilibrium messages from this sender, say G, contain information about the value of D. If G’s equilibrium strategy is to approve a policy change, G(MM), an off-equilibrium message by G, (Q), allows L to infer that the true value of D is 1. Therefore, L’s off-equilibrium beliefs are q1 = 1 and q2 = 0. On the other hand, if G’s equilibrium strategy is to retain the status quo, G(QQ), an off-equilibrium message, (M), allows L to infer that D = 0. L’s off-equilibrium beliefs are q1 = 0 and q2 = 1.
2. If one of the senders has the same preference ordering as L, p1 > sq > p2, off-equilibrium messages from this sender provide L with information about the value of D. For example, if G shares this preference ordering and his equi1ibrium strategy is to approve a policy change G(MM), an off-equilibrium message by G, (Q), allows L to infer that the true value of D is 0. In this case, L’s off-equilibrium beliefs are q1 = 0 and q2 = 1. On the other hand, if G’s equilibrium strategy is G(QQ), L can interpret an off-equilibrium message, (M), to indicate that D = 1. Here, L’s off-equilibrium beliefs are q1 = 1 and q2 = 0.
3. If neither sender shares L’s preference ordering (p1 > sq > p2) and neither sender has a preference ordering opposite of L’s (p2 > sq > p1) G’s and B’s potential off-equilibrium messages provide no new information. L’s off-equilibrium beliefs reflect his priors: q1 = d and q2 = 1 – d.
In the scenario whereby the government has no ex-post authority over the bureaucrat, three types of outcomes exist. First, if either G or B shares L’s preference ordering, the game has separating equilibria. (Recall that L’s preference ordering is fixed: p1 > sq > p2.) In this situation, L is fully informed about the policy consequence and follows the policy prescription of the sender who possesses the same preference ordering. Consequently, the other sender must truthfully reveal the value of D in order to avoid the penalty associated with a legislative veto, regardless of his or her preferences.
In the second and third cases, neither G nor B completely shares L’s preference ordering. In this situation, both separating and pooling equilibria exist. Unfortunately, there is no way to determine which equilibria will be played in these circumstances. In the second case, all three players share a partial preference ordering: either p1 > sq or sq > p2. In equilibrium, both players send the same truthful message about the value of D. L is informed about the consequences of a policy change.
In case 3, pooling equilibria exist if neither sender shares L’s preference ordering. In this case, L gains no new information about the state of the world. Instead, G and B send messages that reflect L’s beliefs about the value of D.
G(MQ), B(EC), L(uudd)
Government: When D = 1, G receives uG(p1) from sending M. Sending Q yields uG(sq). G’s best response is M if uG(p1) > uG(sq). When D = 0, G receives uG(sq) from sending Q. Sending M yields uG(p2). Q is a best response if uG(sq) > uG(p2). Therefore, G(MQ) is a best response to B(EC) and L(uudd) if G has the preference ordering p1 > sq > p2.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(p1) – c. B’s best response is E if uB(p1) > uB(p1) – c. When D = 0, B receives uB(sq) from sending C. Sending E yields uB(sq)–c. B’s best response is C if uB(sq) > uB(sq) – c. Therefore, B(EC) is a best response to G(MQ) and L(uudd) regardless of B’s preferences.
Legislator: If L sees the messages M (from G) and E (from B), L can infer that D = 1. L’s best response in this case is u: uL(p1) > uL(sq). If L receives the messages Q (from G) and C (from B), L can infer that D = 0. L’s best response in this case is d: uL(sq) > uL(p2). Therefore, L(uudd) is a best response to G(MQ) and B(EC) under the assumption that L has the preference ordering p1 > sq > p2.
G(QM), B(EC), L(dduu)
Government: When D = 1, G receives uG(p1) – k from sending Q. Sending M yields uG(sq) – k. G’s best response is Q if uG(p1) – k > uG(sq) – k. When D = 0, G receives uG(sq) – k from sending M. Sending Q yields uG(p2) – k. M, therefore, is a best response if uG(sq) – k > uG(p2) – k. Therefore, G(QM) is a best response to B(EC) and L(dduu) if G has the preference ordering p1 > sq > p2.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(p1) – c. B’s best response is E if uB(p1) > uB(p1) – c. When D = 0, B receives uB(sq) from sending C. Sending E yields uB(sq) – c. B’s best response is C if uB(sq) > uB(sq) – c. Therefore, B(EC) is a best response to G(QM) and L(dduu) regardless ofB’s preferences.
Legislator: If L sees the messages Q (from G) and E (from B), L can infer that D = 1. L’s best response in this case is u: uL(p1) > uL(sq). If L receives the messages M (from G) and C (from B), L can infer that D = 0. L’s best response in this case is d: uL(sq) > uL(p2). Therefore, L(dduu) is a best response to G(QM) and B(EC) under the assumption that L has the preference ordering p1 > sq > p2.
G(MQ), B(EC), L(udud)
Government: When D = 1, G receives uG(p1) from sending M. Sending Q yields uG(p1) – k. G’s best response is M if uG(p1) > uG(p1) – k. When D = 0, G receives uG(sq) from sending Q. Sending M yields uG(sq) – k. G’s best response is Q if uG(sq) > uG(sq) – k. Therefore, G(MQ) is a best response to B(EC) and L(udud) regardless of G’s preferences.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(sq). B’s best response is E if uB(p1) > uB(sq). When D = 0, B receives uB(sq) from sending C. Sending E yields uB(p2). C, therefore, is a best response if uB(sq) > uB(p2). Therefore, B(EC) is a best response to G(MQ) and L(udud) if B has the preference ordering p1 > sq > p2.
Legislator: If L sees the messages M (from G) and E (from B), L can infer that D = 1. L’s best response in this case is u: uL(p1) > uL(sq). If L receives the messages Q (from G) and C (from B), L can infer that D = 0. L’s best response in this case is d: uL(sq) > uL(p2). Therefore, L(udud) is a best response to G(MQ) and B(EC) under the assumption that L has the preference ordering p1 > sq > p2.
G(MQ), B(CE), L(dudu)
Government: When D = 1, G receives uG(p1) from sending M. Sending Q yields uG(p1) – k. G’s best response is M if uG(p1) > uG(p1) – k. When D = 0, G receives uG(sq) from sending Q. Sending M yields uG(sq) – k. G’s best response is Q if uG(sq) > uG(sq) – k. Therefore, G(MQ) is a best response to B(CE) and L(dudu) regardless of G’s preferences.
Bureaucrat: When D = 1, B receives uB(p1) – c from sending C. Sending E yields uB(sq) – c. B’s best response is C if uB(p1) – c > uB(sq)–c. When D = 0, B receives uB(sq) – c from sending E. Sending C yields uB(p2) – c. E, therefore, is a best response if uB(sq) – c > uB(p2) – c. Therefore, B(CE) is a best response to G(MQ) and L(dudu) if B has the preference ordering p1 > sq > p2.
Legislator: If L sees the messages M (from G) and C (from B), L can infer that D = 1. L’s best response in this case is u: uL(p1) > uL(sq). If L receives the messages Q (from G) and E (from B), L can infer that D = 0. L’s best response in this case is d: uL(sq) > uL(p2). Therefore, L(dudu) is a best response to G(MQ) and B(CE) under the assumption that L has the preference ordering p1 > sq > p2.
G(MQ), B(EC), L(uddd)
Government: When D = 1, G receives uG(p1) from sending M. Sending Q yields uG(sq). G’s best response is M if uG(p1) > uG(sq). When D = 0, G receives uG(sq) from sending Q. Sending M yields uG(sq) – k. G’s best response is Q if uG(sq) > uG(sq) – k. Therefore, G(MQ) is a best response to B(EC) and L(uddd) if G has the preference ordering p1 > sq.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(sq). B’s best response is E if uB(p1) > uB(sq). When D = 0, B receives uB(sq) from sending C. Sending E yields uB(sq) – c. C, therefore, is a best response if uB(sq) > uB(sq) – c. Therefore, B(EC) is a best response to G(MQ) and L(uddd) if B has the preference ordering p1 > sq.
Legislator: If L sees the messages M (from G) and E (from B), L can infer that D = 1. L’s best response in this case is u: uL(p1) > uL(sq). If L receives the messages Q (from G) and C (from B), L can infer that D = 0. L’s best response in this case is d: uL(sq) > uL(p2). Therefore, L(uddd) is a best response to G(MQ) and B(EC) under the assumption that L has the preference ordering p1 > sq > p2.
G(MQ), B(EC), L(uuud)
Government: When D = 1, G receives uG(p1) from sending M. Sending Q yields uG(p1) – k. G’s best response is M if uG(p1) > uG(p1) – k. When D = 0, G receives uG(sq) from sending Q. Sending M yields uG(p2). G’s best response is Q if uG(sq) > uG(p2). Therefore, G(MQ) is a best response to B(EC) and L(uuud) if G has the preference ordering sq > p2.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(p1) – c. B’s best response is E if uB(p1) > uB(p1) – c. When D = 0, B receives uB(sq) from sending C. Sending E yields uB(p2). C, therefore, is a best response if uB(sq) > uB(p2) Therefore, B(EC) is a best response to G(MQ) and L(uuud) if B has the preference ordering sq > p2.
Legislator: If L sees the messages M (from G) and E (from B), L can infer that D = 1. L’s best response in this case is u: uL(p1) > uL(sq). If L receives the messages Q (from G) and C (from B), L can infer that D = 0. L’s best response in this case is d: uL(sq) > uL(p2). Therefore, L(uuud) is a best response to G(MQ) and B(EC) under the assumption that L has the preference ordering p1 > sq > p2.
G(MM), B(EE), L(uuuu)
Government: When D = 1, G receives uG(p1) from sending M. Sending Q yields uG(p1) – k. G’s best response is M if uG(p1) > uG(p1) – k. When D = 0, G receives uG(p2) from sending M. Sending Q yields uG(p2) – k. G’s best response is M if uG(p2) > uG(p2) – k. Therefore, G(MM) is a best response to B(EE) and L(uuuu) regardless of G’s preferences.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(p1) – c. B’s best response is E if uB(p1) > uB(p1) – c. When D = 0, B receives uB(p2) from sending E. Sending C yields uB(p2) – c. B’s best response is E if uB(p2) > uB(p2) – c. Therefore, B(EE) is a best response to G(MM) and L(uuuu) regardless of B’s preferences.
Legislator: On equilibrium, where L’s beliefs are q1 = d and q2 = 1 – d, L(uuuu) is a best response if
q1 [uL(p1)] + q2[uL(p2)] ≥ uL(sq)
which implies that
q ≥ [uL(sq) – uL(p2)]/[uL(p1) – uL(p2)].
L’s off-equilibrium beliefs vary across three cases:
1. If either G or B has a preference ordering that is completely opposite L’s ordering, p2 > sq > p1, L’s off-equilibrium beliefs are q1 = 1 and q2 = 0. Therefore, L(uuuu) is a best response if uL(p1) > uL(sq), which is true by assumption.
2. If either G or B has the same preference ordering as L, p1 > sq > p2, L’s off-equilibrium beliefs are q1 = 0 and q2 = 1. Therefore, L(uuuu) is not a best response because L would know to veto a policy change if he observed an off-equilibrium message. In this case, vetoing a policy change would yield u1(sq), which L prefers to uL(p2).
3. If neither G or B shares L’s preference ordering (p1 > sq > p2) and neither G or B has a preference ordering opposite of L’s (p2 > sq > p1), L’s off-equilibrium beliefs reflect his priors: q1 = d and q2 = 1 – d. L(uuuu) is a best response as long as q ≥ [uL(sq) – uL(p2)]/[uL(p1)–uL(p2)].
Therefore, G(MM), B(EE), L(uuuu) is a sequential equilibrium if (1) q ≥ [uL(sq)–uL(p2)]/[uL(p1)–uL(p2)] and (2) neither G or B has the same preference ordering as L: p1 > sq > p2.
G(QQ), B(CC), L(dddd)
Government: When D = 1, G receives uG(sq) from sending Q. Sending M yields uG(sq) – k. G’s best response is Q if uG(sq) > uG(sq) – k. When D = 0, G receives uG(sq) from sending Q. Sending M yields uG(sq) – k. G’s best response is Q if uG(sq) > uG(sq) – k. Therefore, G(QQ) is a best response to B(CC) and L(dddd) regardless of G’s preferences.
Bureaucrat: When D = 1, B receives uB(sq) from sending C. Sending E yields uB(sq) – c. B’s best response is E if uB(sq) > uB(sq) – c. When D = 0, B receives uB(sq) from sending C. Sending E yields uB(sq) – c. B’s best response is E if uB(sq) > uB(sq) – c. Therefore, B(CC) is a best response to G(QQ) and L(dddd) regardless of B’s preferences.
Legislator: On equilibrium, where L’s beliefs are q1 = d and q2 = 1 – d, L(dddd) is a best response if
uL(sq) ≥ q1[uL(p1)] – q2[uL(p2)]
which implies that
q ≤ [uL(sq) – uL(p2)]/[uL(p1)–uL(p2)].
L’s off-equilibrium beliefs vary across three cases:
1. If either G or B has a preference ordering that is completely opposite L’s ordering, p2 > sq > p1, L’s off-equilibrium beliefs are: q1 = 0 and q2 = 1. Therefore, L(dddd) is a best response if uL(sq) > uL(p2), which is true by assumption.
2. If either G or B has the same preference ordering as L, p1 > sq > p2, L’s off-equilibrium beliefs are: q1 = 1 and q2 = 0. Therefore, L(dddd) is not a best response because L would know to approve a policy change if he observed an off-equilibrium message. Approving a policy change would yield uL(p1), which L prefers to uL(sq).
3. If neither G or B shares L’s preference ordering (p1 > sq > p2) and neither G or B has a preference ordering opposite of L’s (p2 > sq > p1) L’s off-equilibrium beliefs reflect his priors: q1 = d and q2 = 1 – d. L(dddd) is a best response as long as q ≤ [uL(sq) – uL(p2)]/[uL(p1)–uL(p2)].
Therefore, G(QQ), B(CC), L(dddd) is a sequential equilibrium if (1) q ≤ [uL(sq)–uL(p2)]/[uL(p1)–uL(p2)] and (2) neither G or B has the same preference ordering as L: p1 > sq > p2.
As in the case with no ex-post authority, separating equilibria exist when G shares L’s preference ordering (case 4). In this scenario, B faces a sanction not only from L but also from G if she deviates.
In all other cases, however, both pooling and separating equilibria exist simultaneously. Unfortunately, the game cannot distinguish when these different equilibria will occur. Case 5 demonstrates that a separating equilibrium exists in this scenario regardless of the preferences of the senders. In case 6, separating equilibria also exist when G and L share a partial preference ordering: P1 > sq or sq > p2. Unlike the scenario with no ex-post authority, however, B’s preferences do not affect the existence of these equilibria.
Finally, if G does not share the same preference ordering with L, pooling equilibria exist (case 7). Unlike the scenario in which G lacked ex-post authority over the bureaucrat, there are two sets of pooling equilibria. The first set, G(MM), B(EE), L(uuuu) and G(QQ), B(CC), L(dddd), is the same as in the first scenario. These equilibria exist if neither G or B has the same preference ordering as L. The second set, G(MM), B(CC), L(udud) and G(QQ), B(EE), L(udud), occurs when B shares the same preference ordering as L. Recall that in the game in which G lacked ex-post authority over the bureaucrat, no pooling equilibria existed if B shared L’s preference ordering.
G(MQ), B(EC), L(uudd)
Government: As in case 1, G(MQ) is a best response to B(EC) and L(uudd) if G has the preference ordering p1 > sq > p2.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(p1) – c – t. B’s best response is E if uB(p1) > uB(p1) – c – t. When D = 0, B receives uB(sq) from sending C. Sending E yields uB(sq) – c – t. B’s best response is C if uB(sq) > uB(sq) – c – t. Therefore, B(EC) is a best response to G(MQ) and L(uudd) regardless of B’s preferences.
Legislator: As in case 1, L(uudd) is a best response to G(MQ) and B(EC) under the assumption that L has the preference ordering p1 > sq > p2.
G(QM)) B(CE)) L(dduu)
Government: As in case 1, G(QM) is a best response to B(CE) and L(dduu) if G has the preference ordering p1 > sq > p2.
Bureaucrat: When D = 1, B receives uB(p1) – k from sending E. Sending C yields uB(p1) – t. B’s best response is E if uB(p1) – k > uB(p1) – t. When D = 0, B receives uB(sq) – k from sending C. Sending E yields uB(sq) – t. B’s best response is C if uB(sq) – k > uB(sq) – t. Therefore, B(EC) is a best response to G(QM) and L(dduu) regardless of B’s preferences and if k < t.
Legislator: As in case 1, L(dduu) is a best response to G(QM) and B(EC) under the assumption that L has the preference ordering p1 > sq > p2.
G(MQ)) B(EC)) L(udud)
Government: When D = 1, G receives uG(p1) from sending M. Sending Q yields uG(p1) – k. G’s best response is M if uG(p1) > uG(p1) – k. When D = 0, G receives uG(sq) from sending Q. Sending M yields uG(sq) – k. Q, therefore, is a best response if uG(sq) > uG(sq) – k. Therefore, G(MQ) is a best response to B(EC) and L(udud) regardless of G’s preference ordering.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(p1) – c. B’s best response is E if uB(p1) > uB(p1) – c. When D = 0, B receives uB(sq) from sending C. Sending E yields uB(sq) – c. C, therefore, is a best response if uB(sq) > uB(sq) – c. Therefore, B(EC) is a best response to G(MQ) and L(udud) regardless of B’s preference ordering.
Legislator: If L sees the messages M (from G) and E (from B), L can infer that D = 1. L’s best response in this case is u: uL(p1) > uL(sq). If L receives the messages Q (from G) and C (from B), L can infer that D = 0. L’S best response in this case is d: uL(sq) > uL(p2). Therefore, L(udud) is a best response to G(MQ) and B(EC) under the assumption that L has the preference ordering p1 > sq > p2.
G(MQ), B(EC), L(uddd)
Government: As in case 2, G(MQ) is a best response to B(EC) and L(uddd) if G has the preference ordering p1 > sq.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(sq) – t. B’s best response is E if uB(p1) > uB(sq) – t. When D = 0, B receives uB(sq) from sending C. Sending E yields uB(sq) – c – t. C, therefore, is a best response if uB(sq) > uB(sq) – c – t. Therefore, B(EC) is a best response to G(MQ) and L(uddd) regardless of B’s preferences.
Legislator: As in case 2, L(uddd) is a best response to G(MQ) and B(EC) under the assumption that L has the preference ordering p1 > sq > p2.
G(MQ), B(EC), L(uuud)
Government: As in case 2, G(MQ) is a best response to B(EC) and L(uuud) if G has the preference ordering sq > p2.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(p1) – c – t. B’s best response is E if uB(p1) > uB(p1) – c – t. When D = 0, B receives uB(sq) from sending C. Sending E yields uB(p2) – t. C, therefore, is a best response if uB(sq) > uB(p2) – t. Therefore, B(EC) is a best response to G(MQ) and L(uuud) regardless of B’s preference ordering.
Legislator: As in case 2, L(uuud) is a best response to G(MQ) and B(EC) under the assumption that L has the preference ordering p1 > sq > p2.
G(MM), B(EE), L(uuuu)
Government: As in case 3, G(MM) is a best response to B(EE) and L(uuuu) regardless of G’s preferences.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(p1) – c – t. B’s best response is E if uB(p1) > uB(p1) – c – t. When D = 0, B receives uB(p2) from sending E. Sending C yields uB(p2) – c – t. B’s best response is E if uB(p2) > uB(p2) – c – t. Therefore, B(EE) is a best response to G(MM) and L(uuuu) regardless of B’s preferences.
Legislator: As in case 3, L(uuuu) is a best response if q ≥ [uL (sq) uL(p2)]/[uL(p1) – uL(p2)]. L’s off-equilibrium beliefs are also the same as in case 3.
Therefore, G(MM), B(EE), L(uuuu) is a sequential equilibrium if (1) q ≥ [uL(sq) – uL(p2)]/[uL(p1) – uL(p2)] and (2) neither G nor B has the same preference ordering as L: p1 > sq > p2.
G(QQ), B(CC), L(dddd)
Government: As in case 3, G(QQ) is a best response to B(CC) and L(dddd) regardless of G’s preferences.
Bureaucrat: When D = 1, B receives uB(sq) from sending C. Sending E yields uB(sq) – c – t. B’s best response is E if uB(sq) > uB(sq) – c – t. When D = 0, B receives uB(sq) from sending C. Sending E yields uB(sq) – c – t. B’s best response is E if uB(sq) > uB(sq) – c – t. Therefore, B(CC) is a best response to G(QQ) and L(dddd) regardless of B’s preferences.
Legislator: As in case 3, L(dddd) is a best response if q ≤ [uL(sq) – uL (p2)]/[uL (p1) – uL (p2)]. L’s off-equilibrium beliefs are also the same as in case 3.
Therefore, G(QQ), B(CC), L(dddd) is a sequential equilibrium if (1) q ≤ [uL(sq) – uL(p2)]/[uL(p1) – uL(p2)] and (2) neither G nor B has the same preference ordering as L: p1 > sq > p2.
G(MM), B(EE), L(udud)
Government: As in case 3, G(MM) is a best response to B(EE) and L(udud) regardless of G’s preferences.
Bureaucrat: When D = 1, B receives uB(p1) from sending E. Sending C yields uB(p1) – c – t. B’s best response is E if uB(p1) > uB(p1) – c – t. When D = 0, B receives uB(p2) from sending E. Sending C yields uB(p2) – c – t. B’s best response is E if uB(p2) > uB(p2) – c – t. Therefore, B(EE) is a best response to G(MM) and L(udud) regardless of B’s preferences.
Legislator: As in case 3, on equilibrium, where L’s beliefs are q1 = d and q2 = 1 – d, L(udud) is a best response if q ≥ [uL(sq) – uL(p2)]/[uL(p1) – uL(p2)].
Although L’s off-equilibrium beliefs are also the same as in case 3, L’s off-equilibrium strategies differ:
1. If G has a preference ordering that is completely opposite L’s ordering, L’s off-equilibrium beliefs are q1 = 1 and q2 = 0. L should approve the policy change because UL(p1) > uL(sq). Therefore, L(udud) is a best response.
If B has a preference ordering completely opposite L’s ordering, L’s off-equilibrium beliefs are q1 = 1 and q2 = 0. Therefore, L(udud) is not a best response because L would prefer to vote u (uL(p1) > uL(sq)) if B sent an off-equilibrium message.
2. If G has the same preference ordering as L, p1 > sq > p2, L’s off-equilibrium beliefs are q1 = 0 and q2 = 1. If G sends an off-equilibrium message, L(udud) is not a best response because L would prefer d to u (uL(sq) > uL(p2))’
If B has the same preference ordering as L, L’s off-equilibrium beliefs are q1 = 0 and q2 = 1. Therefore, L(udud) is a best response because L would know to veto a policy change if he observed an off-equilibrium message from B. In this case, vetoing a policy change would yield uL(sq), which L prefers to uL(p2).
3. If G does not share L’s preference ordering (p1 > sq > p2) and does not have a preference ordering opposite of L’s (p2 > sq > p1), L’S off-equilibrium beliefs reflect his priors: q1 = d and qz = 1 – d. Therefore, L(udud) is a best response as long as q ≥ [uL(sq) – uL(p2)]/[uL(p1) – uL(p2)].
If B does not share L’s preference ordering (p1 > sq > p2) and does not have a preference ordering opposite of L’s (p2 > sq > p1), L’S off-equilibrium beliefs reflect his priors: q1 = d and q2 = 1 – d. In this situation, L(udud) is not a best response to B’s off-equilibrium message. As long as q ≥ [uL(sq) – uL(p2)]/[uL(p1) – uL(p2)], L would prefer to approve the policy than veto it.
Therefore, G(MM), B(EE), L(udud) is a sequential equilibrium if (1) q ≥ [uL(sq) – uL(p2)]/[uL(p1) – uL(p2)], (2) G does not have the same preference ordering as L (p1 > sq > p2), and (3) B has the same preference ordering as L (p1 > sq > p2).
G(QQ), B(CC), L(udud)
Government: As in case 3, G(QQ) is a best response to B(CC) and L(udud) regardless of G’s preferences.
Bureaucrat: When D = 1, B receives uB(sq) from sending C. Sending E yields uB(sq) – c – t. B’s best response is E if uB(sq) > uB(sq) – c – t. When D = 0, B receives uB(sq) from sending C. Sending E yields uB(sq) – c – t. B’s best response is E if uB(sq) > uB(sq) – c – t. Therefore, B(CC) is a best response to G(QQ) and L(udud) regardless of B’s preferences.
Legislator: As in case 3, on equilibrium, where L’s beliefs are q1 = d and q2 = 1 – d, L(udud) is a best response if q ≤ [uL(sq)– uL(p2)]/[uL(p1)–uL(p2)].
Although L’S off-equilibrium beliefs are also the same as in case 3, L’S off-equilibrium strategies differ:
1. If G has a preference ordering that is completely opposite L’s ordering (p2 > sq > p1) L’s off-equilibrium beliefs are q1 = 0 and q2 = 1. L should veto the policy change because uL(sq) > uL(p2). Therefore, L(udud) is a best response.
If B has a preference ordering completely opposite L’s ordering, L’s off-equilibrium beliefs are q1 = 0 and q2 = 1. Therefore, L(udud) is not a best response because L would prefer to vote d (uL(sq) > uL (p2)) if B sent an off-equilibrium message.
2. If G has the same preference ordering as L (p1 > sq > p2), L’s off-equilibrium beliefs are q1 = 1 and q2 = 0. If G sends an off-equilibrium message, L(udud) is not a best response because L would prefer u to d (uL(p1) > uL(sq)).
If B has the same preference ordering as L, L’s off-equilibrium beliefs are q1 = 1 and q2 = 0. Therefore, L(udud) is a best response because L would know to approve a policy change if he observed an off-equilibrium message from B. In this case, approving a policy change would yield uL(p1)’ which L prefers to uL(sq).
3. If G does not share L’s preference ordering (p1 > sq > p2) and does not have a preference ordering opposite of L’s (p2 > sq > p1) L’s off-equilibrium beliefs reflect his priors: q1 = d and q2 = 1 – d. Therefore, L(udud) is a best response as long as q ≤ [uL(sq) – uL(p2)]/[uL(p1) – uL(p2)].
If B does not share L’s preference ordering (p1 > sq > p2) and does not have a preference ordering opposite of L’s (p2 > sq > p1) L’s off-equilibrium beliefs reflect his priors: q1 = d and q2 = 1 – d. In this situation, L(udud) is not a best response to B’s off-equilibrium message. As long as q ≤ [uL(sq) – uL (p2)] – [uL (p1) – uL (p2)], L would prefer to veto the policy than approve it.
Therefore, G(QQ), B(CC), L(udud) is a sequential equilibrium if (1) q ≤ [uL (sq) – uL (p2)] – [uL (p1) – uL(p2)], (2) G does not have the same preference ordering as L (p1 > sq > p2), and (3) B has the same preference ordering as L (p1 > sq > p2).
