This section analyzes the signaling game model by deriving the equilibrium outcomes under different information conditions. It first examines the separating and mixed equilibria, and then develops the corresponding hypotheses based on the strategic behaviors of firms and financial institutions.
In the signaling game framework, a refined Bayesian equilibrium is achieved when the sender’s strategy
and the receiver’s strategy
are jointly consistent with the posterior belief
. Formally, such an equilibrium must satisfy two refinement conditions. (1) Receiver optimality. Given any posterior belief updated after observing a signal
, the receiver (financial institutions and investors) must choose an action that maximizes its expected payoff. (2) Sender optimality. Anticipating the receiver’s optimal response, each type of firm must select a signaling strategy
that maximizes its own expected payoff. These equilibrium requirements can be formally expressed as Formulas (1) and (2):
4.1. Separating Equilibrium
In the signaling game, two potential separating equilibria arise from the sender’s strategy profile: and . Because firms have a clear incentive to apply for green loans at the lower interest rate , the second configuration lacks practical economic relevance. Therefore, we focus on the first separating structure, where firms without substantive green transformation () refrain from greenwashing and apply only for conventional loans at the higher rate , while genuinely green-transforming firms () apply for green loans at . Under this configuration, the signaling game reaches a separating equilibrium in which the sender’s strategy is , and the receiver’s posterior beliefs become and .
- (1)
Receiver’s optimal response
When observing signal
, the financial institution computes the expected payoff from granting the loan (
) and from rejecting the loan (
) as
From Formulas (3) and (4), if , then , meaning that extending the loan yields a higher expected return; hence the optimal action is . Conversely, if , the optimal strategy is not to extend the loan.
When observing signal
, the financial institution’s expected payoff from actions
and
is
Since , the optimal strategy upon receiving is also . Thus, the receiver’s equilibrium strategy is .
- (2)
Sender’s incentive compatibility
Given the receiver’s strategy above, a
firm has no incentive to send the high-interest signal
. Therefore, we only need to compare the payoff of a
firm on the equilibrium path with that off the equilibrium path. The payoff of a
firm when sending
(equilibrium path) is
The payoff of the same firm when deviating to
(off-equilibrium path) is
From Formulas (7) and (8), a firm prefers the equilibrium strategy if . When this condition holds, the expected gain from mimicking a firm is insufficient to offset the additional signaling cost.
Combining the receiver’s optimality condition and the sender’s incentive compatibility condition, a Separating Refined Bayesian Equilibrium exists when the following two conditions are simultaneously satisfied: and . Under these conditions, the strategy profile , constitutes a stable separating equilibrium.
4.2. Mixed Equilibrium
In the mixed-strategy setting, the signal sender may adopt either or . Because firms have no incentive to send the high-interest signal , the analysis focuses exclusively on the second case, in which both and firms choose to send the signal . When firms send with probability , the signaling game converges to a pooling (confounding) equilibrium. In this state, financial institutions update their priors according to Bayes’ rule, yielding the posterior beliefs: and .
- (1)
Receiver’s optimal response
Upon observing the equilibrium-path signal
, the financial institution computes the expected payoff of granting the loan (
) and rejecting the loan (
) as
From Formulas (9) and (10), holds when either and , or simultaneously and . These conditions indicate that when (i) the default-compensation payment received by financial institutions exceeds the losses associated with corporate default, or (ii) the proportion of firms genuinely advancing green transformation is sufficiently large, issuing green loans yields a positive expected return. In such cases, the optimal action for financial institutions is to grant the loan.
When the financial institution observes signal
, the expected payoffs from actions
and
are given by
According to these expressions, requires .
- (2)
Sender’s incentive compatibility
Given the receiver’s action profile
,
firms still have no incentive to deviate by sending signal
. Therefore, the comparison focuses on the payoff of a
firm on the equilibrium path versus that off the equilibrium path. The payoff for a
firm on the equilibrium path (sending
and receiving action
) is:
The payoff for deviating to
when the receiver chooses
is
A firm prefers the equilibrium strategy when . If this condition holds, the firm earns a higher pay off by sending the pooling signal rather than deviating.
Combining the receiver’s optimality condition and the sender’s incentive compatibility condition, a Mixed Refined Bayesian Equilibrium exists when the following two conditions are simultaneously satisfied: and . Under these conditions, the strategy profile , constitutes a stable mixed equilibrium.
4.3. Hypothesis Development
The equilibrium analysis above provides the theoretical basis for deriving empirically testable hypotheses. The signaling game framework does not simply assume that green M&A conveys positive information; rather, it clarifies the conditions under which such information can become credible, be recognized by financial institutions, and be transmitted to capital market valuation. In particular, the model highlights the roles of signaling costs, imitation incentives, default risk, financing terms, compensation arrangements, and financial institutions’ belief updating in shaping the effectiveness of green M&A signals. Rather than serving as a full structural-estimation framework, these theoretical elements provide a mechanism-based foundation for translating the equilibrium implications into observable empirical predictions.
Specifically, the model implies three core empirical relationships. First, if green M&A serves as a credible signal of firms’ green transformation commitment, it should be reflected in capital market valuation. Second, if financial institutions recognize and respond to this signal, green M&A should affect green credit allocation. Third, if green credit reflects financial institutions’ recognition and resource-allocation response, it may transmit part of the value effect of green M&A. Accordingly, H1–H3 examine the valuation effect of green M&A, the green credit response, and the transmission role of green credit, respectively.
(1) Under the separating equilibrium, firms transmit financing signals that truthfully reflect their underlying transformation intentions. Specifically,
firms actively engaged in green transition send
signals to apply for low-interest green loans, while
firms with weak transition incentives send
signals and can only obtain conventional loans at higher interest rates. At this stage, the profits earned by
firms undertaking green M&A and sending
signals, denoted as
, and the profits earned by
firms sending
signals, denoted as
, yield a benefit differential
, expressed as
In Equation (17), the first two components, the loan interest rate spread and the market-attention premium , are both positive. The sign of the third component depends on firms’ default-related gains or losses and therefore requires separate discussion. In the extreme case of , reduces to two positive components, ensuring > 0. When , the benefits recovered through default include the loan principal and interest, while the costs include the compensation payment , credit losses , and foregone credit gains . In practice, both and are sufficiently large to make default gains extremely small. Thus, even if the third term is negative, its magnitude remains limited, and the overall remains positive.
These results imply that firms actively engaging in green M&A achieve higher returns regardless of their default risk level. Green M&A enables access to low-interest green credit, reducing transformation financing costs and improving investment returns [
55]. It also serves as a credible strategic signal that enhances market trust, stimulates positive investor expectations, and brings additional long-term value such as brand premiums [
10,
56]. Therefore, we propose the following hypothesis:
H1: Green M&A effectively increase corporate market value.
(2) When the conditions and are simultaneously satisfied, the signaling game reaches a separating refined Bayesian equilibrium . This equilibrium has two implications. First, if the combined benefits from interest rate spreads and market-attention gains do not offset the cost of information embellishment , firms have no incentive to disguise their true transformation level. Second, if the compensation mechanism fully covers lenders’ losses from corporate default, financial institutions can profit under both strategies, implying a Pareto-improving outcome.
When conditions and are met, the game reaches a mixed refined Bayesian equilibrium . In this case, firms lacking substantive transformation incentives ( firms) may still engage in greenwashing to obtain low-cost credit. Financial institutions’ optimal responses then depend on the proportion of truly transforming companies. When is sufficiently high, a mixed equilibrium emerges.
In practice, lenders often recover losses through guarantees, collateral disposal, or restructuring [
52]. At the same time, more firms use green M&A as a channel for strategic upgrading under low-carbon transition pressures [
21]. These realities make both separating and mixed equilibria empirically plausible. Thus, green M&A functions as a credible signal that influences the allocation of credit resources. Accordingly, the following hypothesis is proposed:
H2: Green M&A serve as a signaling mechanism for credit resource allocation.
(3) Under both separating and mixed equilibria,
firms have no incentive to send
signals and will always choose
signals to access lower-interest green credit. In the separating equilibrium,
firms with weak transformation incentives must send
signals and can only obtain conventional loans. This reflects their lack of substantive green actions and reliance on higher-cost credit for short-term support. In the mixed equilibrium, however,
firms may choose to embellish information to obtain green credit. The profits and their differences between these two scenarios, when
firms send
under separation and
under mixed, are given by
The results show that interest rate spreads
and market-attention gains
are the key drivers of firms’ incentives to greenwash. These factors determine whether the equilibrium shifts from a separating equilibrium to a pooling equilibrium. In other words, credit resource allocation plays a central mediating role, influencing both firms’ financing decisions and market outcomes. Green credit provides low-cost funding for green investment, improving project returns and supporting long-term sustainable development [
18]. Positive market responses to green investment further boost firm valuation [
28,
34]. Therefore, the following hypothesis is proposed:
H3: Green credit plays an important mediating role in enhancing corporate market value through green M&A.
The above hypotheses capture the core transmission chain implied by the signaling game framework, namely that green M&A signals may affect capital market valuation through financial institutions’ recognition and green credit allocation. However, the model also implies that this transmission process is conditional rather than automatic. The effectiveness of green M&A signals depends on the credibility of the signal source, the visibility of the signal during transmission, the stability of the external interpretive environment, as well as firm-level conditions related to signaling costs and default risk. Therefore, in addition to testing H1–H3, this study further conducts moderating-effect and heterogeneity analyses to examine the boundary conditions under which green M&A signals are strengthened or weakened.