Contract withdrawals and equilibrium in competitive markets with adverse selection

Abstract

In competitive common value adverse selection markets, existence of a pure strategy equilibrium is often justified by appealing to Wilson’s (J Econ Theory 16(2):167–207, 1977) concept of ‘anticipatory equilibrium.’ The anticipatory equilibrium is based on the notion that all market participants expect unprofitable contracts to be withdrawn. We present a model of individual contract withdrawals that captures the strategic process underlying the anticipatory equilibrium concept: We introduce an additional—endogenously ending—stage into the Rothschild and Stiglitz (Q J Econ 90(4):629–649, 1976) model in which initial contracts can be withdrawn repeatedly after observation of competitors’ contract offers and withdrawals. Individual contract withdrawal allows for a rich strategic interaction. We show that an equilibrium exists where consumers obtain their respective second-best efficient Miyazaki–Wilson–Spence (MWS) contracts. However, this equilibrium requires latent contracts on offer. Furthermore, any individually rational and incentive-compatible allocation that earns nonnegative profits on aggregate can be sustained as equilibrium allocation. We further allow for contract addition as in Riley’s (Econometrica 47(2):331–359, 1979) ‘reactive equilibrium.’ Allowing for contract addition does not change the set of possible outcomes.

Appendix: Proofs

1.1 Notations

Let $\varOmega _\mathrm{MWS}:= \left\{ \omega ^{H}_\mathrm{MWS},\omega ^{L}_\mathrm{MWS}\right\}$ denote the set of MWS contracts, $\varOmega _{RS}:= \left\{ \omega ^{H}_{RS},\omega ^{L}_{RS} \right\}$ the set of RS contracts, and

$$\begin{aligned}&\varOmega _\mathrm{LR}:=\big \{\omega \in \varOmega |u^{L}(\omega ) \le u^{L}(\omega ^{L}_\mathrm{MWS}), u^{L}(\omega ) \ge u^{L}(\omega ^{L}_{RS}) \nonumber \\&\qquad \text { and } (1-p^{L})(w_{01}-w_{1})+p^{L}(w_{02}-w_{2})=0 \big \} \end{aligned}$$

the (compact) set of contracts that lie on the L-type fair insurance line and yield an L-type an expected utility that is not lower than expected utility from her RS contract but not higher than her expected utility from her MWS contract. Note that, of course, $\omega ^{L}_{RS} \in \varOmega _\mathrm{LR}$.

Furthermore, let

$$\begin{aligned}&\varOmega _{CS}:= \{\omega \in \varOmega |u^{L}(\omega )> u^{L}(\omega ^{L}_\mathrm{MWS}), u^{H}(\omega ) < u^{H}(\omega ^{L}_\mathrm{MWS}) \nonumber \\&\qquad \text { and }\; (1-p^{L})(w_{01}-w_{1})+p^{L}(w_{02}-w_{2})> 0\} \end{aligned}$$

denote the cream-skimming contract set with respect to the MWS contracts. We denote the virtual profit of firm f if stage 2 would end after $t-1$ by

$$\begin{aligned} \pi ^{f} (\varOmega _{t-1}):= & {} \text { } \gamma [(1-p^{H})(w_{01}-\bar{w}^{H}_{1, t-1})+p^{H}(w_{02}-\bar{w}^{H}_{2,t-1})](1/\bar{k}^H(\varOmega _{t-1}))\\&{\mathbb {1}}_{\{\bar{\omega }_{t-1}^{H} \in \text { } \varOmega _{t-1}^{f}\}}+\,(1- \gamma )[(1-p^{L})(w_{01}-\bar{w}^{L}_{1,t-1})\nonumber \\&+\,p^{L}(w_{02}-\bar{w}^{L}_{2,t-1})](1/\bar{k}^L(\varOmega _{t-1})){\mathbb {1}}_{\{\bar{\omega }_{t-1}^{L} \in \text { } \varOmega _{t-1}^{f}\}}, \end{aligned}$$

where ${\mathbb {1}}$ is an indicator function. Similarly, we denote by

$$\begin{aligned} \pi ^{f,H}(\varOmega _{t-1})= & {} \gamma [(1-p^{H})(w_{01}-\bar{w}^{H}_{1,t-1})\\&+\,p^{H}(w_{02}-\bar{w}^{H}_{2,t-1})](1/\bar{k}^H(\varOmega _{t-1})){\mathbb {1}}_{\{\bar{\omega }_{t-1}^{H} \in \text { } \varOmega _{t-1}^{f}\}} \end{aligned}$$

and

$$\begin{aligned} \pi ^{f,L}(\varOmega _{t-1})= & {} (1-\gamma )[(1-p^{L})(w_{01}-\bar{w}^{L}_{1,t-1})+p^{L}(w_{02}\\&-\,\bar{w}^{L}_{2,t-1})](1/\bar{k}^L(\varOmega _{t-1})){\mathbb {1}}_{\{\bar{\omega }_{t-1}^{L} \in \text { }\varOmega _{t-1}^{f}\}} \end{aligned}$$

the virtual profits on H types and L types, respectively.

Furthermore, let $\Delta _{t}^{{\setminus } LR} := \Delta _{t} {\setminus } \varOmega _\mathrm{LR}$ denote the set of contracts offered in t excluding the LR contract set. We denote by $\ddot{\omega }_{t}^{J}=(\ddot{w}_{1,t}^{J},\ddot{w}_{2,t}^{J})$ the contract such that

$$\begin{aligned} \ddot{\omega }_{t}^{J}\displaystyle \in \arg \max _{\omega \in \Delta _{t}^{{\setminus } LR} } \text { }u^{J}(\omega ) \end{aligned}$$

and

$$\begin{aligned} \displaystyle \ddot{w}^J_{2,t} \ge \tilde{w}^J_{2,t} \text { } \forall \text { } \tilde{\omega }^J_t \in \arg \max _{\omega \in \Delta _{t}^{{\setminus } LR} } \text { }u^{J}(\omega ) \end{aligned}$$

$\ddot{\omega }_{t}^{J}$ is the highest coverage contract maximizing J-type utility among the contracts offered in t when the LR contract set is removed. Note that ‘above’ MWS and ‘below’ RS contracts, $\ddot{\omega }_{t}^{J}$ and $\bar{\omega }_{t}^{J}$ coincide. Then, let $\hat{\varOmega }_{t-1}^{f,LR}:=\{ \omega \in \varOmega _{t-1}^f | u^{H}(\omega ) \le u^{H} (\ddot{\omega }_{t-1}^{H})\}$. $\hat{\varOmega }_{t-1}^{f,LR}$ is the set where all contracts are removed from $\varOmega _{t-1}^f$ which are more attractive for the H types than the best contract, outside the set $\varOmega _\mathrm{LR}$, currently on offer.

Lastly, to incorporate no insurance in a simple manner, we impose that $(w_{01},w_{02}) \in \varOmega ^{f}_{t}$ for all $f\in F$ and $t \ge 0$. To ease exposition, we will not explicitly add $(w_{01},w_{02})$ as part of stage 1 and stage 2 contract menus in the following proofs, but just assume that no insurance is ‘offered’ by each firm in stage 1, and in every round in stage 2, alongside the stated contract menus.

1.2 Proof of Lemma 1

Assume that an equilibrium with the MWS allocation, i.e., an equilibrium in which each customer of type J receives her respective MWS contract in stage 3, exists without latent contracts. Then, for all $f \in F$ and t, $\varOmega ^{f}_{t} \subseteq \varOmega _\mathrm{MWS}$, and every $f \in F$ makes at most zero profits. We will show that a profitable deviation exists. Assume firm $\bar{f}$ deviates and offers a contract menu $\{ \omega _{cs}, \omega _p \}$ in stage 1, where $\omega _{cs} \in \varOmega _{CS}$ and $\omega _p$ makes a strictly positive profit if taken out by the entire population but is not preferred by high risks to their MWS contract. Since the contract sets of each firm are finite, stage 2 ends for sure after some $\hat{t} < \infty$. Then if $\varOmega _\mathrm{MWS} \cap \Delta _{\hat{t}} \ne \emptyset$, then the deviator makes a strictly positive profit with low risks by keeping $\omega _{cs}$ on offer for all $t \le \hat{t}$. If $\varOmega _\mathrm{MWS} \cap \Delta _{\hat{t}} = \emptyset$ , then there exists $\tilde{t}$ such that $\varOmega _\mathrm{MWS} \cap \Delta _{\tilde{t}-1} \ne \emptyset$ and $\varOmega _\mathrm{MWS} \cap \Delta _{\tilde{t}}= \emptyset$. Then, $\bar{f}$ makes a strictly positive profit by choosing $\varOmega ^{\bar{f}} = \{ \omega _{cs}, \omega _p \}$ for all $t \le \tilde{t}$ and withdrawing $\omega _{cs}$ in $\tilde{t}+1$ while keeping $\omega _p$ on offer.

1.3 Proof of Proposition 1

1.3.1 Equilibrium strategy

The equilibrium strategy of firms is the following: In stage 1, firms $f \in F{\setminus } \{n-1,n\}$ offer $\varOmega ^{f}_{0}=\varOmega _\mathrm{MWS} \cup \varOmega _{RS} \cup \varOmega _\mathrm{LR}$. Firms $n-1$ and n offer the LR contracts, i.e., $\varOmega ^{n-1}_{0}=\varOmega ^{n}_{0}=\varOmega _\mathrm{LR}$. In stage 2, we fully describe the strategy at all subgames when in stage 1 at most one firm did offer a contract set different from those specified above. If at most one firm $\bar{f} \ne f$ deviated in stage 1, the strategy of firm $f \in F$ in round $t>0$ specifies

$$\begin{aligned} \alpha ^{f}_t (h_t)= \left\{ \begin{array}{l@{\quad }l} \varOmega ^{f}_{t-1} &{} \mathrm{if}\; \pi ^{f}(\varOmega _{t-1}) \ge 0;\\ \varOmega ^{f}_{t-1}{\setminus } \varOmega _\mathrm{MWS} &{} \mathrm{if}\,\pi ^{f}(\varOmega _{t-1})< 0\, \mathrm{and}\, \varOmega ^{f}_{t-1} \cap \varOmega _\mathrm{MWS} \ne \emptyset ;\\ \hat{\varOmega }_{t-1}^{f,LR} &{} \mathrm{if}\;\pi ^{f}(\varOmega _{t-1})< 0\, \mathrm{and}\, \varOmega ^{f}_{t-1} \cap \varOmega _\mathrm{MWS} = \emptyset .\\ \end{array} \right. \end{aligned}$$

Note that the stage 2 strategy is the same for all firms, i.e., including firms $n-1$ and n who only offer the LR contracts in stage 1. The strategy specifies that if firms make a nonnegative virtual profit after round $t-1$, e.g., on MWS contracts or on RS contracts, they do not withdraw any contracts in round t. If a firm makes a virtual loss after round $t-1$, and the firm still offers a MWS contract, the firm withdraws the MWS contract menu. If a firm makes virtual losses and does not offer any MWS contract, then the firm withdraws contracts such that the remaining (compact) contract set does not contain any contract that is preferred by high risks to the best high-risk contract when LR contracts are assumed not on offer, but leaves exactly the contract that cream-skims low risks with respect to this contract. Observe that the strategy specifies that if there is one firm offering a contract set in stage 1 that is different from $\varOmega _\mathrm{MWS} \cup \varOmega _{RS} \cup \varOmega _\mathrm{LR}$, both RS contracts always remain on offer. Observe also that a firm always withdraws a non-empty set of contracts if it makes a virtual loss.

1.3.2 No profitable deviation

If all firms follow the above strategy, no firm withdraws any contract in $t=1$ and stage 2 ends after $t=1$, firms make zero expected profit, and a customer of type J receives her J-type MWS contract.

We will first show that there is no profitable deviation. We will then show that the strategy profile/equilibrium is subgame perfect (at the withdrawal stage). To show that there is no profitable deviation we proceed in two steps: First, we show that a deviator serves some H types. Second, we show that if the deviator serves some H types, she cannot be making a strictly positive profit.

Consider a deviating firm $\bar{f}$ that offers $\varOmega _{0}^{\bar{f}}$ in stage 1 and has a strategy $\hat{\alpha }^{\bar{f}}: \text { } h_t \longmapsto \varOmega ^{\bar{f}}_{t}$ in stage 2. Let $\varOmega _{\hat{t}}^{f}$ be the final set of contract offers of firm f, i.e., $\varOmega _{\hat{t}}^{f}=\varOmega _{\hat{t}-1}^{f}$$\forall$$f\in F$. Then it must be that $\pi ^{f}(\varOmega _{\hat{t}-1})\ge 0$$\forall$$f\in F{\setminus } \left\{ \bar{f} \right\}$ as otherwise, according to the equilibrium strategy defined above, a firm $f\in F{\setminus } \left\{ \bar{f} \right\}$ would withdraw a non-empty set of contracts in $\hat{t}$ and $\hat{t}$ would not be the final round in stage 2.

Now assume $\pi ^{\bar{f}}(\varOmega _{\hat{t}}) > 0$. As $\pi ^{f}(\varOmega _{\hat{t}-1})\ge 0$$\forall$$f\in F{\setminus } \left\{ \bar{f} \right\}$, we will show that $\bar{w}_{\hat{t}}^{H} \in \varOmega _{\hat{t}}^{\bar{f}}$: Since $\pi ^{\bar{f}}(\varOmega _{\hat{t}}) > 0$, $\bar{f}$ serves some customers. To show that it cannot be possible that $\bar{f}$ serves only L types, assume on the contrary that $\bar{f}$ only serves L types. If L types prefer an insurance contract to remaining uninsured, then H types prefer to be insured as well. As $\bar{f}$ only serves L types, then at least one firm $f\in F{\setminus } \left\{ \bar{f} \right\}$ serves H types and the share of L types among customers at f is less than $1-\gamma$. There are three possible cases:

Case 1$\bar{\omega }_{\hat{t}}^{H}=\omega ^H_\mathrm{MWS}$. Now any firm $f\in F{\setminus } \left\{ \bar{f} \right\}$ that serves H types with $\omega ^H_\mathrm{MWS}$ and has a share of L types among customers that is less than $1-\gamma$ does not make a nonnegative profit.²⁴ This contradicts $\pi ^{f}(\varOmega _{\hat{t}-1})\ge 0$$\forall$$f\in F{\setminus } \left\{ \bar{f} \right\}$.

Case 2$\bar{\omega }_{\hat{t}}^{H} \in \varOmega _\mathrm{LR}$. Any contract $\omega \in \varOmega _\mathrm{LR}$ if taken up by some H types is loss making, independent of whether it is also taken up by some L types. This contradicts $\pi _{f}(\varOmega _{\hat{t}-1})\ge 0$$\forall$$f\in F{\setminus } \left\{ \bar{f} \right\}$.

Case 3$\bar{\omega }_{\hat{t}}^{H}=\omega ^H_{RS}$. Then, from the strategy of firms $n-1$ and n, at least one firm still has the L-type RS contract on offer when $\bar{\omega }_{\hat{t}}^{H}=\omega ^H_{RS}$. Then, there is no contract (menu) that $\bar{f}$ can offer attracting L types only and making a positive profit, which is a contradiction.

Hence, $\bar{\omega }_{\hat{t}}^{H} \in \varOmega _{\hat{t}}^{\bar{f}}$. We will now show that if $\bar{\omega }_{\hat{t}}^{H} \in \varOmega _{\hat{t}}^{\bar{f}}$, $\bar{f}$ cannot be making a positive profit. First, note that the RS contracts remain always on offer. This is because firms $n-1$ and n do not withdraw the low-risk-type RS contract as long as the high-risk-type RS contract is on offer, and all other firms do not withdraw the high-risk-type RS contract as long as the low-risk-type RS contract is on offer. Then, it follows that $u^H (\bar{\omega }_{\hat{t}}^{H}) \ge u^H (\omega ^H_{RS})$. There are again three possible cases:

Case 1$u^H (\bar{\omega }_{\hat{t}}^{H}) \ge u^H (\omega ^H_\mathrm{MWS})$. Then, $\omega ^L_\mathrm{MWS}$ will not have been withdrawn by any firm $f\in F{\setminus } \left\{ \bar{f} \right\}$. As $\omega ^L_\mathrm{MWS}$ is on offer from firms $f\in F{\setminus } \left\{ \bar{f} \right\}$, by construction of the MWS contracts, $\pi ^{\bar{f}}(\varOmega _{\hat{t}}) \le 0$ for the cases that $\bar{f}$ only serves H types or both types.

Case 2$u^H (\bar{\omega }_{\hat{t}}^{H}) < u^H (\omega ^H_\mathrm{MWS})$ and $\bar{\omega }_{\hat{t}}^{L} \in \varOmega _{CS}$, i.e., the preferred L-type contract is in the cream-skimming contract set with respect to MWS contracts. Hence, both MWS contracts are not on offer at any firm $f\in F{\setminus } \left\{ \bar{f} \right\}$ as they have been withdrawn, since firms make losses on MWS contracts if a cream-skimming contract is on offer. Thus, any firm $f\in F{\setminus } \left\{ \bar{f} \right\}$ does not serve neither L types nor H types since it does not offer any contract $w \in \varOmega _{CS}$. This implies that with $\bar{\omega }_{\hat{t}}^{H}$ the deviator firm $\bar{f}$ serves all H types. However, by construction of the MWS contracts, there is no incentive-compatible menu of contracts with $\bar{w}_{\hat{t}}^{L} \in \varOmega _{CS}$ that is profit making; hence, $\pi _{\bar{f}}(\varOmega ^{\hat{t}}) < 0$.

Case 3$u^H (\bar{\omega }_{\hat{t}}^{H}) < u^H (\omega ^H_\mathrm{MWS})$ and $\bar{\omega }_{\hat{t}}^{L} \notin \varOmega _{CS}$. First consider that $\bar{\omega }_{\hat{t}}^{H} \in \varOmega _\mathrm{LR} \cup \varOmega _\mathrm{RLR}$ with

$$\begin{aligned} \varOmega _\mathrm{RLR}:= & {} \{ \omega \in \varOmega | u^{L}(\omega )\le u^{L}(\omega ^{L}_\mathrm{MWS}) \text { and } u^{H}(\omega )\ge u^{H}(\omega ^{H}_{RS}); \\&(1-p_{L})(w_{01}-w_{1})+p_{L}(w_{02}-w_{2})]< 0 \}, \end{aligned}$$

i.e., $\bar{\omega }_{\hat{t}}^{H}$ is such that it is making zero or negative profits even when taken out by L types. Then it immediately follows $\pi ^{\bar{f}}(\varOmega _{\hat{t}}) < 0$. Therefore, consider $\bar{\omega }_{\hat{t}}^{H} \notin \varOmega _\mathrm{LR} \cup \varOmega _\mathrm{RLR}$. From the strategy of any firm $f\in F{\setminus } \left\{ \bar{f} \right\}$, $\bar{\omega }_{\hat{t}}^{L} \in \varOmega _{\hat{t}}^{f}$$\forall$$f \in F{\setminus } \left\{ \bar{f} \right\}$ as in every t firms $f \in F{\setminus } \left\{ \bar{f} \right\}$ leave the largest (compact) subset from $\varOmega _\mathrm{LR}$, including the L-type RS contract, on offer that attracts low risks but does not give high risks a higher expected utility than that of the most preferred contract of H types from $\Delta _t {\setminus } \varOmega _\mathrm{LR}$. Then, either the deviator’s contract for the low risks is loss making, or the share of low-risk customers at $\bar{f}$ is lower than $1-\gamma$, while $\bar{f}$ serves all high risks such that $\pi ^{\bar{f}}(\varOmega _{\hat{t}}) \le 0$.

Hence, $\pi ^{\bar{f}}(\varOmega _{\hat{t}}) \le 0$ which is a contradiction.

1.3.3 Subgame perfection

When in stage 1 not more than one firm offered a contract set different from those specified above for firms $n-1$ and n and all other firms, respectively, then the strategy prescribes that in t, firms do not withdraw any contract if they make a nonnegative virtual profit, but always withdraw the virtually loss-making contracts in t if they make a virtual loss. Following a stage 1 contract menu $\varOmega _\mathrm{MWS} \cup \varOmega _{RS} \cup \varOmega _\mathrm{LR}$ for firms $f \in F {\setminus } \{n-1,n\}$ and $\varOmega _\mathrm{LR}$ for firms $n-1$ and n, this strategy profile is subgame perfect for the following reason: First consider firms $f \in F {\setminus } \{n-1,n\}$. If a firm were to makes a virtual loss, it withdraws the loss-making contract(s). Except for the MWS and RS high-risk-type contract, none of the offered contracts is profit making if taken out by the whole population. Now the strategy profile is such that a firm $f \in F {\setminus } \{n-1,n\}$ can never make a positive profit on either of the two contracts: As long as the MWS high-risk contract is on offer either the low-risk MWS or the cream-skimming LR contracts are still on offer as well by at least one firm, $n-1$ or n, thus low risks will not buy the high-risk MWS contract. The high-risk RS contract will also never be bought by the whole population, as the low-risk RS contract remains always on offer by at least firm, $n-1$ or n. There are two firms offering only the LR contracts to ensure that the strategies of all firms are subgame perfect even if there is a stage 1 deviation, i.e., not offering only the LR contracts, by either firm $n-1$ and n. Having two firms ensures that there is at least one firm offering only the LR contracts such that withdrawal of contract by firms $f \in F{\setminus } \{n-1,n\}$ is subgame perfect. For firms $n-1$ and n, since they only offer the LR contracts and can thus not make a strictly positive profit on any contract, it is subgame perfect to leave non-loss-making LR contracts, including the low-risk RS contract, on offer.

We have specified that in stage 2, at all subgames reached when in stage 1 more than one firm did offer a contract set different from $\varOmega _\mathrm{MWS} \cup \varOmega _{RS} \cup \varOmega _\mathrm{LR}$, firms play any subgame perfect equilibrium strategy of the subgame starting in stage 2. Thus, the strategy profile is subgame perfect for these subgames by definition.

1.4 Proof of Proposition 2

(i) We construct the corresponding equilibria for any allocation $\phi =(\omega ^H_{\phi },\omega ^L_{\phi }) \in \varPhi$.

Case I: $\phi$ such that $u^H (\omega ^H_{\phi }) \le u^H (\omega ^H_\mathrm{MWS})$ and $\omega ^L_{\phi }$ makes nonnegative profits with low risks.

First, we consider the case that $\phi$ is such that the high-risk contract does not give high risks a higher expected utility than the high-risk MWS contract and there is no cross-subsidization from high to low risks. Among these, there are allocations that give high risks a higher expected utility than their RS contract such that the allocation is cross-subsidizing from low to high risks, and some where the high risks receive an expected utility that is equal or lower than that from their RS contract.

We will construct equilibria in the following way: To prevent stage 1 deviations, some firms, which we will henceforth call inactive MWS firms, offer exactly the MWS, RS and LR contracts. For these it will be credible not to withdraw MWS contracts if they observe a stage 1 deviation, but they will withdraw their offered contract sets along the equilibrium path. To prevent stage 2 deviations, e.g., by inactive MWS firms that try to pool all risks on the high-risk RS contract, or by a firm offering the intended equilibrium allocation but withdrawing a contract from this menu in order to make a higher profit, there will be two further firms, which we will henceforth call again LR firms. The LR firms offer the LR contracts, as in proof of Proposition of 1, and in addition, a contract that is zero-profit-making when taken out by low risks gives low risks a lower expected utility than the low-risk contract from $\phi$ but a higher expected utility than the high-risk contract from $\phi$, but gives high risks a lower expected utility than the high-risk contract from $\phi$. That is, when only the intended equilibrium allocation is on offer and the low-risk contract from the intended equilibrium allocation would be withdrawn, this contract would be taken out by low risks and only low risks, and would not make losses. Last, we let exactly one firm offer the intended equilibrium allocation. This is to prevent, in a simple manner, the problem of withdrawal of a loss-making high-risk contract by some firm when the intended equilibrium allocation is cross-subsidizing and offered by more than one firm.

1.4.1 Equilibrium strategies

We specify the strategy in stage 1, and from stage 2 onward if in stage 1 at most one player offers a contract set different from equilibrium contracts.²⁵ Let $\varOmega _{IA}:=\varOmega _\mathrm{MWS} \cup \varOmega _{RS} \cup \varOmega _\mathrm{LR}.$ Furthermore, let $\omega ^{\hat{L}_{\phi }}$ denote the contract that solves:

$$\begin{aligned}&\displaystyle \max _{\omega } \text { }u^{L}(\omega )\end{aligned}$$

(6)

$$\begin{aligned}&\hbox {s.t.}\nonumber \\&u^{H}(\omega ^{H}_{\phi })\ge u^{H}(\omega )\end{aligned}$$

(7)

$$\begin{aligned}&u^{L}(\omega ^{L}_{\phi })\ge u^{L}(\omega )\end{aligned}$$

(8)

$$\begin{aligned}&(1-p^{L})(w_{01}-w_{1})+p^{L}(w_{02}-w_{2})= 0. \end{aligned}$$

(9)

The equilibrium strategies of firms are the following: In stage 1, firm 1 sets $\varOmega ^{1}_{0}=\{\omega ^H_{\phi },\omega ^L_{\phi }\}$, firms $f \in F{\setminus } \{1, n-1,n\}$ offer $\varOmega ^{f}_{0}=\varOmega _{IA}$, and firms $n-1$ and n set $\varOmega ^{n-1}_{0}=\varOmega ^{n}_{0}=\varOmega _\mathrm{LR} \cup \{\omega ^{\hat{L}_{\phi }}\}$.²⁶ We will henceforth call all firms $f \in F{\setminus } \{1,n-1,n\}$ inactive MWS firms. We will call firms $n-1$ and n LR firms. The inactive MWS firms withdraw their contract menus on the equilibrium path; the LR firms withdraw some, but not all contracts on the equilibrium path.²⁷

To ease notation, we will denote histories at the beginning of stage 2 in the following way: We say ND1 holds if every firm offered the contract menus in stage 1 as specified by the equilibrium strategy above, i.e., $\varOmega ^{1}_{0}=\{\omega ^H_{\phi },\omega ^L_{\phi }\}$, $\varOmega ^{f}_{0}=\varOmega _{IA}$ for all firms $f \in F{\setminus } \{1, n-1,n\}$ and $\varOmega ^{n-1}_{0}=\varOmega ^{n}_{0}=\varOmega _\mathrm{LR} \cup \{\omega ^{\hat{L}_{\phi }}\}$ for firms $n-1$ and n. We say D1 holds if some firm did offer a contract menu in stage 1 different from the contract menu specified by the equilibrium strategy above, i.e., either $\varOmega ^{1}_{0} \ne \{\omega ^H_{\phi },\omega ^L_{\phi }\}$, or $\varOmega ^{f}_{0} \ne \varOmega _{IA}$ for some firm $f \in F{\setminus } \{1, n-1,n\}$ or $\varOmega ^{n-1}_{0} \ne \varOmega _\mathrm{LR} \cup \{\omega ^{\hat{L}_{\varPhi }}\}$ or $\varOmega ^{n}_{0}\ne \varOmega _\mathrm{LR} \cup \{\omega ^{\hat{L}_{\phi }}\}$.

If at most one firm did offer a contract set different from those specified above in stage 1, the strategy of firms $f \in F{\setminus } \{1, n-1,n\}$ in round $t>0$ in stage 2 specifies

$$\begin{aligned} \alpha ^{f}_t (h_t)= \left\{ \begin{array}{l@{\quad }l} \varOmega ^{f}_{t-1} &{} \mathrm{if}\, D1\, \mathrm{holds}\,\mathrm{and}\, \pi ^{f}(\varOmega _{t-1}) \ge 0;\\ \varOmega ^{f}_{t-1}{\setminus } \varOmega _\mathrm{MWS} &{} \mathrm{if}\, D1\,\mathrm{holds,}\, \pi ^{f}(\varOmega _{t-1})< 0\, \mathrm{and}\, \varOmega ^{f}_{t-1} \cap \varOmega _\mathrm{MWS} \ne \emptyset ;\\ \hat{\varOmega }_{t-1}^{f,LR} &{} \mathrm{if}\, D1\,\mathrm{holds,}\, \pi ^{f}(\varOmega _{t-1})< 0\, \mathrm{and}\, \varOmega ^{f}_{t-1} \cap \varOmega _\mathrm{MWS} = \emptyset ;\\ \{(w_{01},w_{02})\} &{} \mathrm{if}\, ND1\,\mathrm{holds}. \end{array} \right. \end{aligned}$$

The first three lines describe the strategy if in the first stage someone deviated by offering a different set of contracts, the last contract withdrawal along the equilibrium path.

If at most one firm did offer a contract set different from those specified above in stage 1, the strategy of firms $n-1$ and n in round $t>0$ in stage 2 specifies

$$\begin{aligned} \alpha ^{f}_t (h_t)= \left\{ \begin{array}{l@{\quad }l} \varOmega ^{f}_{t-1} &{} \mathrm{if}\,\pi ^{f}(\varOmega _{t-1}) \ge 0;\\ \hat{\varOmega }_{t-1}^{f,LR} &{} \mathrm{if}\,\pi ^{f}(\varOmega _{t-1})< 0\, \mathrm{and}\, \omega ^H_{\phi } \in \Delta _{t-1};\\ \hat{\varOmega }_{t-1}^{f,LR} {\setminus } \{\omega ^{\hat{L}_{\phi }}\} &{} \mathrm{if}\,\pi ^{f}(\varOmega _{t-1})< 0\, \mathrm{and}\,\omega ^H_{\phi } \notin \Delta _{t-1}.\\ \end{array} \right. \end{aligned}$$

Note that the inactive LR firms always leave the contract $\omega ^{\hat{L}_{\phi }}$ on offer as long as $\omega ^H_{\phi }$ is on offer. Note further that as long as the high-risk RS contract is on offer, the inactive LR firms also keep the low-risk RS contract on offer (from their LR contracts).

Lastly, the strategy of firm 1 in round $t>0$ in stage 2 specifies

$$\begin{aligned} \alpha ^{1}_t (h_t)= \left\{ \begin{array}{l@{\quad }l} \varOmega ^{1}_{t-1} &{} \mathrm{if}\, \pi ^{1}(\varOmega _{t-1}) \ge 0;\\ \varOmega ^{1}_{t-1}{\setminus } \{\underline{\omega }_{\phi }\} &{} \mathrm{if}\, \pi ^{1}(\varOmega _{t-1}) < 0,\\ \end{array} \right. \end{aligned}$$

where $\underline{\omega }_{\phi }$ denotes the contract from $\varOmega ^f_{t-1}$ on which the virtual losses would be made. In stage 2, at all subgames reached when in stage 1 more than one firm did offer a contract set different from those specified above, firms play any subgame perfect equilibrium strategy of the subgame starting in stage 2.

The above strategies prescribe the following: Firm 1 offers the intended equilibrium allocation as contract menu. Inactive MWS firms offer the MWS, RS and LR contracts. LR firms offer the LR contracts and $\omega ^{\hat{L}_{\phi }}$. If a firm deviates in stage 1 by offering a different contract menu than specified, then inactive MWS firms behave as in the proof of Proposition 1.

If every firm offers contract menus in stage 1 according to the equilibrium strategy, the inactive MWS firms withdraw their contract menus in the first round of stage 2.²⁸ Thus, the logic is that inactive MWS firms prevent stage 1 deviations. Contract menus of LR firms are constructed to prevent stage 2 deviations. First, LR contracts are left on offer by inactive LR firms such that inactive MWS firms cannot profitably deviate by not withdrawing the high-risk MWS or RS contract. Second, $\omega ^{\hat{L}_{\varPhi }}$ is left on offer as long as the high-risk contract from the intended equilibrium allocation is on offer, such that firm 1 cannot profitably deviate by withdrawing the low-risk contract from the intended equilibrium allocation. Firm 1, which in stage 1 only offered as contract set the equilibrium allocation, leaves the allocation on offer if it makes nonnegative virtual profits, and withdraws the respective loss-making contract(s) if they make virtual losses. This latter part is to ensure subgame perfection.

If all firms follow the above respective strategies, the inactive MWS firms offering the MWS, RS and LR contracts withdraw their contract menus in $t=1$, LR firms withdraw all contracts from the LR contracts that give high risks a higher utility than $\omega ^H_{\phi }$ in $t=2$, firm 1 does not withdraw any contract and stage 2 ends after $t=3$. Firm 1 makes nonnegative profits and a customer of type J receives her J-type contract from $\phi$.²⁹

1.4.2 No profitable deviation

If there is a stage 1 deviation, inactive firms act as in proof of Proposition 1. The proof proceeds along the same lines as the proof of Proposition 1 and is therefore omitted. Thus, it remains to show that there is no profitable deviation in stage 2 if all firms offer the contract menus in stage 1 as specified in the equilibrium strategy. First, consider a stage 2 deviation by some firm $\bar{f} \in F{\setminus }\{1, n-1,n\}$, i.e., an inactive MWS firm. In stage 2, $\bar{f}$ can only profitably deviate if it either pools all risks on the high-risk MWS contract or on the high-risk RS contract. By the equilibrium strategy of firms the inactive LR firms $n-1$ and n, pooling all risks on either the high-risk MWS or RS contract cannot occur, since the corresponding cream-skimming contracts attracting low risks will not be withdrawn by firms $n-1$ and n as long as the high-risk MWS or, respectively, low-risk RS contracts are still on offer and firms $n-1$ and n thereby do not make losses on these low-risk contracts. Next, consider a stage 2 deviation by firm 1. Firm 1 cannot profitably deviate by not offering a cross-subsidized high-risk contract if $\phi$ is cross-subsidizing high risks: Since it is the only firm offering $\{\omega ^H_{\phi },\omega ^L_{\phi }\}$, withdrawing the cross-subsidized contract and pooling all risks on the low-risk contract always yields equal or lower profits than keeping all contracts from $\{\omega ^H_{\phi },\omega ^L_{\phi }\}$ on offer. Second, firm 1 cannot profitably deviate by withdrawing $\omega ^L_{\phi }$ and keeping $\omega ^H_{\phi }$ on offer, i.e., pooling all risks on $\omega ^H_{\phi }$. This is because as long as $\omega ^H_{\phi }$ is on offer, the inactive LR firms keep $\omega ^{\hat{L}_{\phi }}$ on offer which attracts low risks if $\omega ^L_{\phi }$ is withdrawn. Lastly, no contract offered by the inactive LR firms is potentially profit making such that there is no profitable stage 2 deviation for these firms.

1.4.3 Subgame perfection

Note that inactive MWS firms are indifferent between being inactive or making zero profits on MWS contracts. Since $n \ge 5$, i.e., there are at least two firms offering only MWS, RS and LR contracts, and there are two inactive LR firms offering only the LR contracts and $\omega ^{\hat{L}_{\phi }}$, it is indeed subgame perfect to keep MWS contracts on offer if a stage 1 deviation, by either firm 1, another inactive firm or an inactive LR firm is observed. For this type of deviation (stage 1 deviation), withdrawal prescribed by the equilibrium strategy is subgame perfect as shown in the proof of Proposition 1.³⁰

Now consider stage 2. First we check subgame perfection of the strategies of inactive MWS firms. The strategy prescribes complete withdrawal if there was no stage 1 deviation. Since, with inactive LR firms offering the LR contracts as long as they are not making a virtual loss with those, there is no contract on which positive profits can be made by inactive MWS firms with another withdrawal strategy in the continuation game. Then, it is subgame perfect to withdraw the complete contract menu in $t=1$ if there is no stage 1 deviation. Next, we check the strategy of firm 1 in the withdrawal phase. Since firm 1 is the only firm offering $\{\omega ^H_{\phi },\omega ^L_{\phi }\}$, it is subgame perfect not to withdraw $\omega ^H_{\phi }$ as long as $\omega ^L_{\phi }$ is on offer. This is because the profit of a firm offering any incentive-compatible menu would be lower if it were to withdraw the high-risk contract and pool all risks on the low-risk contract. Second, it is subgame perfect not to withdraw $\omega ^L_{\phi }$ since the inactive LR firms offer $\omega ^{\hat{L}_{\phi }}$ as long as $\omega ^H_{\phi }$ is on offer and would attract the low risks such that firm 1 cannot pool all risks on $\omega ^H_{\phi }$. Thus, it is subgame perfect to leave all contracts from $\{\omega ^H_{\phi },\omega ^L_{\phi }\}$ on offer. Last, we check strategies in the withdrawal phase of inactive LR firms. They cannot make a positive profit on any of their offered contracts and withdraw loss-making contracts from LR. Furthermore, they withdraw $\omega ^{\hat{L}_{\phi }}$ if $\omega ^H_{\phi }$ is not on offer and $\omega ^{\hat{L}_{\phi }}$ is potentially loss making. Thus, the strategy is subgame perfect in the withdrawal stage.

There are two special cases, where the equilibrium strategies as specified above do not work. One is where the high risks obtain a utility larger than the utility at the MWS contract. One example for such a scenario would be pooling at full insurance at the average fair risk premium. The other is where the allocation is such that the high-risk contract makes a profit, while the low-risk contract makes a loss.

Case II:$\phi$such that$u^H (\omega ^H_{\phi }) > u^H (\omega ^H_\mathrm{MWS})$.

Consider the case that $\phi$ is such that the high-risk contract gives high risks a higher expected utility than the high-risk MWS contract.³¹ In that case, the inactive MWS firms might make positive (virtual) profits with their low-risk-type MWS contract. In order to make it optimal for the inactive firms to withdraw the low-risk-type MWS contract, the contract set of firm 1 needs to be amended. If it contains a continuum of contracts, firm 1 can make the ‘threat’ to withdraw contracts indefinitely if the other firms do not withdraw their low-risk-type MWS contract.³² Thus, let $\varOmega _\mathrm{LRS}$ denote the contracts on the low-risk fair insurance line between the low-risk RS contract and the no insurance point, i.e., the line segment between these two points,³³ and let $\varOmega ^f_{LRS_{t}}:=\{\omega \in \varOmega _\mathrm{LRS}|\omega \in \varOmega ^f_{t}\}$, i.e., the set of contracts from the line segment $\varOmega _\mathrm{LRS}$ that firm f has on offer in t.

Furthermore, in order to slightly adjust stage 1 offers of inactive LR firms and keep their stage 2 strategy simple, let $\varOmega _{\underline{LR}}:=\{\omega \in \varOmega _\mathrm{LR}| u^L (\omega ) \le u^L (\omega ^L_{\phi })\}$. Note that $\omega ^{\hat{L}_{\phi }} \in \varOmega _{\underline{LR}}$. We specify the strategy in stage 1, and from stage 2 onward if in stage 1 at most one player offers a contract set different from equilibrium contracts. The equilibrium strategies of firms are the following: In stage 1, firm 1 sets $\varOmega ^{1}_{0}=\{\omega ^H_{\phi },\omega ^L_{\phi }\} \cup \varOmega _\mathrm{LRS}$, firms $f \in F{\setminus } \{1, n-1,n\}$ offer $\varOmega ^{f}_{0}=\varOmega _{IA}$ and firms $n-1$ and n set $\varOmega ^{n-1}_{0}=\varOmega ^{n}_{0}=\varOmega _{\underline{LR}}$. If at most one firm did offer a contract set different from those specified above in stage 1, the strategy of firms $f \in F{\setminus } \{1\}$ in round $t>0$ in stage 2 are the same as specified under Case I.

As discussed above, there is a difference in the stage 2 strategy of firm 1, the strategy of firm 1 in round $t>0$ in stage 2 specifies

$$\begin{aligned} \alpha ^{1}_t (h_t)= \left\{ \begin{array}{l@{\quad }l} \varOmega ^{1}_{t-1} &{} \mathrm{if}\,\pi ^{1}(\varOmega _{t-1}) \ge 0;\\ \varOmega ^{1}_{t-1}{\setminus } \hat{\varOmega }_{LRS_{t-1}} &{} \mathrm{if}\, \pi ^{1}(\varOmega _{t-1}) < 0,\\ \end{array} \right. \end{aligned}$$

where $\hat{\varOmega }_{LRS_{t-1}}$ denotes the open line segment from the (closed) line segment $\varOmega ^{1}_{LRS_{t-1}}$ whose length is 1 / 2 of the length of $\varOmega ^{1}_{LRS_{t-1}}$ and whose endpoints have the lowest distance to the low-risk RS contract.

In stage 2, at all subgames reached when in stage 1 more than one firm did offer a contract set different from those specified above, firms play any subgame perfect equilibrium strategy of the subgame starting in stage 2.

The above strategies prescribe the following: Firm 1 offers the intended equilibrium allocation as contract menu and additionally the continuum of contracts on the line segment between the low-risk RS contract and the no-insurance contract. As long as firm 1 makes virtual losses with its contract menu, it withdraws some contracts on the line segment. Thus, the game would go on indefinitely if firm 1 does not make nonnegative virtual profits with its contract menu. Note that firm 1 never withdraws $\{\omega ^H_{\phi },\omega ^L_{\phi }\}$. The inactive MWS firms offer the RS, MWS and LR contracts as above and inactive LR firms offer the LR contracts that do now give low risks an expected utility higher than that from $\omega ^L_{\phi }$. If there is no stage 1 deviation, then inactive MWS firms withdraw their contract menus. If a stage 1 deviation is observed, then inactive MWS firms have the same strategy as firms in the proof of Proposition 1. Furthermore, if there is a stage 2 deviation where inactive firms do not withdraw the low-risk MWS contract such that firm 1 would make a loss if stage 2 ended, in each t firm 1 withdraws a (open) non-empty set of contracts from the line segment $\varOmega ^{1}_{LRS_{t-1}}$ such that stage 2 does not end as long as the low-risk MWS contract is still on offer from an inactive firm.

If all firms follow the above respective strategies, inactive MWS firms withdraw their contract menus in $t=1$, firm 1 withdraws an open line segment from the LRS contracts in $t=1$, inactive LR firms do not withdraw any contract in $t=1$, there is no contract withdrawal in $t=2$ and stage 2 ends after $t=2$. All firms except firm 1 do not sell any contracts, firm 1 makes nonnegative profits and a customer of type J receives her J-type contract from $\phi$.

1.4.4 No profitable deviation

If there is a stage 1 deviation, inactive MWS firms play as in proof of Proposition 1. Firm 1 withdraws a non-empty set of contracts in every t if it makes virtual losses at the end of $t-1$. Thus, either there is a stage 1 deviation by firm 1, and the proof proceeds along the same lines as proof of Proposition 1 and is therefore omitted, or there is a stage 1 deviation by any other firm. Then, however, either firm 1 makes a virtual loss and stage 2 does not end, or firm 1 does not make a loss. Then, since firm 1 leaves $\{\omega ^H_{\phi },\omega ^L_{\phi }\}$ always on offer and does not make a loss, there is no contract menu that, if offered alongside $\{\omega ^H_{\phi },\omega ^L_{\phi }\}$, makes a positive profit such that there is no profitable deviation.

It remains to show that there is no profitable stage 2 deviation. First, consider a stage 2 deviation by an inactive MWS firm. Since firm 1 withdraws a non-empty set of contracts in every t when the low-risk MWS contract was still on offer at the end of $t-1$, stage 2 does not end, and firms receive zero profits if the low-risk MWS contract remains on offer. Thus, no inactive firm can profitably deviate by leaving the low-risk MWS contract on offer. Furthermore, an inactive MWS firm cannot deviate by pooling all risks on the high-risk MWS or RS contract. This is since either firm 1 makes a virtual loss (and stage 2 does not end), or firm 1 does not make a virtual loss, but then 1 serves all high and low risks, which contradicts an inactive MWS firm pooling risks on the high-risk MWS or RS contract. Next, consider a stage 2 deviation by firm 1. As in Case I, firm 1 cannot profitably deviate by withdrawing $\omega ^L_{\phi }$, since then low risks would be attracted by $\omega ^{\hat{L}}_{\phi }$ offered by the inactive LR firms. Inactive LR firms cannot profitably deviate in stage 2 with the same reasoning as in Case I.

1.4.5 Subgame perfection

Note that all firms except firm 1 are indifferent between being inactive, making zero profits on MWS contracts and making zero profits because stage 2 does not end. We will only go through subgame perfection for the subgames that differ from those in Proposition 1 and Case I above. For inactive MWS firms, it is subgame perfect to withdraw the low-risk MWS contract and with it the complete contract menu, as otherwise they make zero profits since stage 2 does not end. For firm 1, it is subgame perfect not to let stage 2 end as long as it makes a negative profit.

Case III:$\phi$such that$\omega ^L_{\phi }$makes losses with low risks.

With a loss-making low-risk contract, a problem is that any firm offering $\{\omega ^H_{\phi },\omega ^L_{\phi }\}$ would like to withdraw this loss-making low-risk contract, either to pool all risks on the high-risk contract or not serve the low risks, both of which would yield higher profits.³⁴ Whereas pooling the low risks on the profitable high-risk contract can be prevented in a similar way as in Case I above, not serving the low risks with the loss-making contract cannot be prevented by latent contracts, as the low-risk contract is already loss making. However, the withdrawal of the low-risk contract can be prevented by amending the contract set of some other firm, e.g., an inactive LR firm. If it contains a continuum of contracts, this firm can make the ‘threat’ to withdraw contracts indefinitely if firm 1 withdraws $\omega ^L_{\phi }$.³⁵ Thus, let $\varOmega _{NH}$ denote the contracts on the full insurance line between, and including, $(w_{02}, w_{02})$ and $(w_{01}-(r^H +\epsilon ), w_{01}-(r^H +\epsilon ))$, where $r^H$ denotes the high risk’s risk premium, i.e., the closed line segment between these two points.³⁶ Let $\varOmega ^f_{NH_{t}}:=\{\omega \in \varOmega _{NH}|\omega \in \varOmega ^f_{t}\}$, i.e., the set of contracts from the line segment $\varOmega _{NH}$ that firm f has on offer in t.

We specify the strategy in stage 1, and from stage 2 onward if in stage 1 at most one player offers a contract set different from equilibrium contracts. The equilibrium strategies of firms are the following: In stage 1, firm 1 sets $\varOmega ^{1}_{0}=\{\omega ^H_{\phi },\omega ^L_{\phi }\}$, firms $f \in F{\setminus } \{1, n-1,n\}$ offer $\varOmega ^{f}_{0}=\varOmega _{IA}$, firm $n-1$ sets $\varOmega ^{n-1}_{0}=\varOmega _\mathrm{LR}$, and firm n sets $\varOmega ^{n}_{0}=\varOmega _\mathrm{LR} \cup \varOmega _{NH}$. If at most one firm did offer a contract set different from those specified above in stage 1, the strategy of firms $f \in F{\setminus } \{n\}$ in round $t>0$ in stage 2 is the same as specified under Case I.

The strategy of firm n in round $t>0$ in stage 2 specifies

$$\begin{aligned} \alpha ^{n}_t (h_t)= \left\{ \begin{array}{l@{\quad }l} \varOmega ^{n}_{t-1} &{} \mathrm{if}\, \omega ^L_{\phi } \in \Delta _{t-1}\, \mathrm{and}\, \pi ^{n}(\varOmega _{t-1}) \ge 0;\\ \hat{\varOmega }_{t-1}^{n,LR} &{} \mathrm{if}\,\omega ^L_{\phi } \in \Delta _{t-1}\, \mathrm{and}\, \pi ^{n}(\varOmega _{t-1})< 0;\\ \varOmega ^{n}_{t-1}{\setminus } \hat{\varOmega }_{NH_{t-1}} &{} \mathrm{if}\,\omega ^L_{\phi } \notin \Delta _{t-1},\\ \end{array} \right. \end{aligned}$$

where $\hat{\varOmega }_{NH_{t-1}}$ denotes the open line segment from the (closed) line segment $\varOmega ^{n}_{NH_{t-1}}$ whose length is 1 / 2 of the length of $\varOmega ^{n}_{NH_{t-1}}$ and whose endpoints have the lowest distance to the point $(w_{02}, w_{02})$.

In stage 2, at all subgames reached when in stage 1 more than one firm did offer a contract set different from those specified above, firms play any subgame perfect equilibrium strategy of the subgame starting in stage 2.

The above strategy prescribes the following: Firm n offers the LR contracts and additionally a continuum of contracts on the full insurance line that yield high risks (and low risks) lower expected utility than remaining uninsured. If $\omega ^L_{\phi } \notin \Delta _{t-1}$, firm n withdraws a non-empty set of contracts in every t, i.e., if $\omega ^L_{\phi }$ is not offered in stage 1 or withdrawn in some t, stage 2 does not end and all firms make zero profits.

If there is no stage 1 deviation, then inactive MWS firms withdraw their contract menus. If a stage 1 deviation is observed, then inactive MWS firms have the same strategy as firms in the proof of Proposition 1. If there is a stage 2 deviation by inactive MWS firms, firms $n-1$ and n do not withdraw some cream-skimming contracts. If there is a stage 2 deviation by firm 1 in the form of withdrawal of $\omega ^L_{\phi }$, stage 2 does not end.

If all firms follow the above respective strategies, inactive MWS firms withdraw their contract menus in $t=1$, firms $n-1$ and n withdraw their LR contracts in $t=2$, firm 1 does not withdraw any contract and stage 2 end after $t=3$. All firms except firm 1 do not sell any contracts, firm 1 makes nonnegative profits and a customer of type J receives her J-type contract from $\phi$.

1.4.6 No profitable deviation

If there is a stage 1 deviation, inactive MWS firms play as in proof of Proposition 1. The proof proceeds along the same lines as proof of Proposition 1 and is therefore omitted.

It remains to show that there is no profitable stage 2 deviation. There is no profitable stage 2 deviation for an inactive MWS firm or for the inactive LR firms $n-1$ and n following the same reasoning as in Case I. Last, consider a stage 2 deviation by firm 1. The only potentially profitable deviation is withdrawal of $\omega ^L_{\phi }$. Then, however, from the strategy of firm n, stage 2 does not end, and firm 1 receives zero profits such that the deviation is not profitable.

1.4.7 Subgame perfection

We only go through the part that differs from Case I. Note that firm n can never make a positive profit and is indifferent between being inactive, making zero profits on low risks with some LR contract and making zero profits because stage 2 does not end. Then, it is subgame perfect for firm n not to let stage 2 end if $\omega ^L_{\phi }$ is withdrawn.

(ii) We show that any allocation that is not an element of $\varPhi$ cannot be an equilibrium allocation. An allocation that is not an element of $\varPhi$ is not individually rational for consumers, incentive-compatible or loss making. If the allocation is not individually rational or incentive-compatible, some type does not buy its intended contract. A loss-making allocation cannot be an equilibrium allocation as the number of firms $n < \infty$ such that some firm f that makes negative profits would withdraw contracts.

1.5 Proof of Proposition 3

Assume to the contrary that an equilibrium with an allocation $(\bar{w}^{H},\bar{w}^{L}) \ne (\omega ^H_{WMS}, \omega ^L_{WMS})$, where no contract was withdrawn on the equilibrium path, exists. We will show that there is a profitable deviation in stage 1 and hence a contradiction.

Since it is an equilibrium, each firm makes a nonnegative expected profit. Since $(\bar{w}^{H},\bar{w}^{L})$ is the equilibrium allocation and no contract is withdrawn on the equilibrium path, there does not exist a contract $\omega \in \mathop {\bigcup }\nolimits _{f \in F} \text { } \varOmega ^f_{0}$ with $u^{J}(\omega ) > u^J (\bar{w}^{J})$.

Now since $(\bar{w}^{H},\bar{w}^{L}) \ne (\omega ^H_{WMS}, \omega ^L_{WMS})$ there exist some firm $\tilde{f}$ and incentive-compatible contracts $\tilde{\omega }^H$ and $\tilde{\omega }^L$ with $u^{L}(\tilde{\omega }^L)>u^{L}(\bar{w}^{L})$ such that if $\varOmega ^{\tilde{f}}_{\hat{t}}=\left\{ \tilde{\omega }^H,\tilde{\omega }^L \right\}$ and all consumers buy their respective contract at firm $\tilde{f}$ in stage 3, the profit of firm $\tilde{f}$ would be higher than the profit of firm $\tilde{f}$ when the equilibrium allocation is $(\bar{w}^{H},\bar{w}^{L})$, see, for example, proof of Lemma C1 in Diasakos and Koufopoulos (2015). But then, for some firm $\tilde{f}$, setting $\varOmega ^{\hat{f}}_{t}=\{\tilde{\omega }^H, \tilde{\omega }^L\}$ for all t is a profitable deviation: First, since there is no contract $\omega \in \mathop {\bigcup }\nolimits _{f \in F{\setminus } \tilde{f}} \text { } \varOmega ^f_{0}$ with $u^{J}(\omega ) > u^J (\bar{w}^{J})$, and since $u^{L}(\tilde{\omega }^L)>u^{L}(\bar{w}^{L})$, withdrawal of some contract(s) in some round t by some other firm cannot induce L types to not buy $\tilde{\omega }^L$ in stage 3 since $\tilde{\omega }^L$ always remains the best contract on offer for L types. Second, keeping $\{\tilde{\omega }^H, \tilde{\omega }^L\}$ on offer for all t yields firm $\tilde{f}$ higher profits than when the equilibrium allocation is $(\bar{w}^{H},\bar{w}^{L})$ and thus nonnegative profits such that the menu will not be withdrawn by firm $\tilde{f}$. Hence, there is a profitable deviation and thus a contradiction.

1.6 Proof of Proposition 4

‘$\Leftarrow$’ Fix an equilibrium in the original game with equilibrium strategy $\alpha ^{*f}_t (h_t)$. Then consider the following strategy $\bar{\alpha }^{f}_t(h_t)$ in the extended game: Firm $f \in F$ offers $\varOmega _0^{*f}$ in stage 1. In stage 2, round $t=1$, the strategy specifies $\bar{\alpha }^{f}_t(h_t)=\alpha ^{*f}_t(h_t)$. For $t\ge 2$, $\bar{\alpha }^{f}_t(h_t)=\alpha ^{*f}_t(h_t)$ if $\varOmega ^j_{t-1} \subseteq \varOmega ^j_{t-2}$$\forall$$j \in F$. That is, firms play as in the original game as long as no contract was added by any firm in the previous round for $t\ge 2$. If, however, any firm $\hat{f} \ne f$ adds a contract in $t-1$, i.e., $\varOmega ^j_{t-1} \nsubseteq \varOmega ^j_{t-2}$ for some $j \in F {\setminus } \{f\}$, then $\bar{\alpha }^{f}_t(h_t)=\varOmega _0^{*f}$. That is, if some firm adds a contract in some $t-1$, then all other firms reoffer stage 1 contract menus in t.

Now firstly, this strategy yields the same equilibrium allocation as in the original game as on the equilibrium path, each firm $f \in F$ takes the same action in stage 1 and in all rounds of stage 2 as on the corresponding equilibrium path in the original game.

It remains to show that there is no profitable deviation. A profitable deviation here means a deviation such that profits are higher than in equilibrium in the original game. There are two cases to consider:

Case 1 Stage 2 does not end.

If stage 2 does not end, all firms make zero profits. Then a deviation is not profitable.

Case 2 Stage 2 ends.

Then, there exists a largest t where a deviator adds a contract, denoted by $\tilde{t}$. According to the above strategy, all non-deviating firms will then offer their initial set of contracts in $\tilde{t}+1$. Then, however, there is no profitable deviation from the proof of Proposition 2.

‘$\Rightarrow$’ We prove this by contraposition. An allocation that is not an equilibrium allocation in the withdrawal game is not individually rational for consumers, incentive-compatible or loss making. If the allocation is not individually rational or incentive-compatible, some type does not take her intended equilibrium contract. A loss-making allocation cannot be an equilibrium allocation as the number of firms $n < \infty$ such that some firm f that makes negative profits would withdraw loss-making contract(s).