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Purification and disambiguation of Ellsberg equilibria

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Abstract

Recently, the use of Knightian (uncertain) strategies in normal form games has received increasing attention. The use of uncertain acts in games leads to new (Ellsberg) equilibria. We provide a foundation of the new equilibrium concept in the spirit of Harsanyi by proving an extension of the Purification Theorem for \(2\times 2\) normal form games. Our result implies that Ellsberg equilibria are limits of equilibria in slightly perturbed games with private information. In such equilibria, players use pure or maxmin strategies only.

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Notes

  1. In terms of payoffs, \(p^*\ne \bar{p}\) and \(q^*\ne \bar{q}\) are, respectively, equivalent to

    $$\begin{aligned} \frac{\pi _2^4-\pi _2^3}{\pi _2^4-\pi _2^3+\pi _2^1-\pi _2^2}\ne \frac{\pi _1^4-\pi _1^3}{\pi _1^4-\pi _1^3+\pi _1^1-\pi _1^2} \qquad \text {and} \qquad \frac{\pi _1^4-\pi _1^2}{\pi _1^4-\pi _1^2+\pi _1^1-\pi _1^3}\ne \frac{\pi _2^4-\pi _2^2}{\pi _2^4-\pi _2^2+\pi _2^1-\pi _2^3}. \end{aligned}$$
  2. Obviously, for any game \(\varGamma \) of class II there is a corresponding game of class III obtained from \(\varGamma \) by exchanging the two players’ strategies and payoffs.

  3. For our later results on purification and disambiguation, the common knowledge assumption is not needed. We just use the fact that each player faces ambiguity about the other player’s behavior.

  4. The intuition for this can be grasped by comparing Figs. 8, 9b–d presented below. The graph of \(U_1\big (.,q_1,\epsilon r\big )\) has the same intercept as the graph of \(U_1\big (.,q_1)\) shown in Fig. 8. The slopes of these two graphs differ because of the additional term \(p\epsilon r\).

  5. Game \(\varGamma _2\) belongs to the set of games Riedel and Sass (2013) cover if one exchanges the two columns.

  6. See again Lemma 8 for a proof.

  7. The expression for \(q_{\max }\) is obtained by simplifying Eq. (17) in Lemma 6. The second expression is valid when \(t^*\) has a positive value, which is verified for small \(\epsilon \).

  8. Equations (14) to (17) correspond to the case for which all thresholds belong to the support. If it was not the case, the expression for these integrals should be modified. Any threshold outside the support must be replaced by the nearest point in the support. These modifications are necessary for Eqs. (14) to (17) to correspond to Eq. (7).

  9. The Ellsberg profile induced by the equilibrium in the perturbed game changes with \(\epsilon \) and, therefore, so do the endpoints defining the thresholds’ value. Nevertheless, if the endpoints change little, the intuition carries over.

  10. As shown in the proof of Lemma 7, the endpoints of the interval \(\big [p^-(\epsilon ),p^+(\epsilon )\big ]\) exist.

  11. For quasi-proper Ellsberg equilibria, the endpoints of one player are equal and those equilibria belong to two of the above-defined types. The mixed strategy equilibrium \((p^*,q^*)\) belongs to all four.

  12. See proof of Lemma 12.

  13. This statement holds as well when \(k_r=1\) or \(k_t=1\) as \(p_{\max } \in (A,B)\) and \(q_{\max } \in (C,D)\), implying, respectively, that \(p_{\min } \in \{A,B\}\) or \(q_{\min } \in \{C,D\}\).

  14. Weak inequalities \(p_L^l\le p_U^l\) and \(p_L^u\le p_U^u\) come from the case \(k_r=1\). For such value of \(k_r\), we have \(p_L^l= p_U^l=A\) and \(p_L^u= p_U^u=B\), as shown in the proof of Lemma 11.

  15. The strict monotonicity of \(h_p\) and \(h_q\) is only valid as long as \(k_r<1\) and \(k_t<1\). When \(k_r=1\) (\(k_t=1\)), function \(h_p\) (\(h_q\)) is not injective. This is not a problem for our purpose as these functions are injective and surjective on a smaller domain. For example, when \(k_r=1\), function \(h_p\) is bijective on \([\hat{r},1]\subset [-1,1]\) defined in the proof of Lemma 11. The definition of probabilities \(p^-(\epsilon )\) and \(p^+(\epsilon )\) must be adapted such that \(h_r\) has the appropriate domain of image \([\hat{r},1]\). On this basis, a similar mapping can be constructed.

  16. Function \(p_{\min }\) is strictly decreasing in \(r^*\) when \(k_r<1\).

  17. In Lemma 14, the conditions under which the expression \(max\{r',r''\}\) is used for \(r^*\) are such that the solution p of the implicit equation \(U_1(p,0)=U_1(p,1)\) is non-positive. The conditions under which the expression \(min\{r',r''\}\) is used for \(r^*\) are such that the solution p of the implicit equation \(U_1(p,0)=U_1(p,1)\) is equal to or larger than 1.

  18. If \(q_{\min }=q_{\max }\) then \(r^*=r'=r''\). But if \(q_{\min }<q_{\max }\), two cases can arise: either \(U_1(p,q_{\min })<U_1(p,q_{\max })\) for all \(p \in (0,1]\) or \(U_1(p,q_{\min })>U_1(p,q_{\max })\) for all \(p \in (0,1]\). Assume that the game \(\varGamma \) is such that the first of these two cases arises. The relevant threshold \(r^*\) is therefore \(r'\) associated with \(q_{\min }\). Under the payoff conditions leading to \(r^*=max\{r',r''\}\), \(U_1(p,q_{\min })\) and \(U_1(p,q_{\max })\) are two straight lines which cross at \(\tilde{p}\le 0\). As a result, we have \(U_1(0,q_{\min })-U_1(1,q_{\min })>U_1(0,q_{\max })-U_1(1,q_{\max })\). The other case leads to the same conclusion.

  19. For all \(\varGamma \in {{\varvec{\Gamma }}}\) such that Player 2’s payoffs are Row Dominant, those conditions are obtained for pure strategies of Player 2 by replacing \(\pi _1^1\) by \(\pi _2^1\), \(\pi _1^4\) by \(\pi _2^4\), \(\pi _1^2\) by \(\pi _2^3\), and \(\pi _1^3\) by \(\pi _2^2\).

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Acknowledgements

We are grateful to Claude d’Aspremont, Jan-Henrik Steg, Francois Maniquet, John Weymark, Igor Muraviev, Nikoleta Ŝćekić and Martin Van der Linden for the comments provided on previous versions of this document.

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Correspondence to Benoit Decerf.

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Financial Support through Grant DFG-ANR Ri 1128-6-1 and the Belgian Fond National de la Recherche Scientifique (mandat d’aspirant FC 95720) is gratefully acknowledged.

Appendix

Appendix

1.1 Examples of games for classes II and III

We provide examples of games for classes II and III. First, \(\varGamma _3 \in {{\varvec{\Gamma }}}^{II}\) is defined in the left panel of Fig. 10. This game is such that \(p^*=\frac{1}{2}\ne 0= \bar{p}\) and \(q^*=\frac{2}{3}\ne \frac{1}{2}= \bar{q}\). Second, \(\varGamma _4 \in {{\varvec{\Gamma }}}^{III}\) is defined in the right panel of Fig. 10. This game is such that \(p^*=\frac{2}{3}\ne \frac{1}{2}= \bar{p}\) and \(q^*=\frac{1}{2}\ne 0= \bar{q}\).

Fig. 10
figure 10

Normal form of games \(\varGamma _3\) (on the left) and \(\varGamma _4\) (on the right), respectively, belonging to classes II and III

1.2 Proof of Lemma 1

Take any basic game \(\varGamma \in {{\varvec{\Gamma }}}\) and any \(\epsilon >0\). We only show that if \(\bar{p} \in (0,1)\), then we have

$$\begin{aligned} U_1\big (\bar{p},[q_1,q_2],\epsilon r\big )=&~U_1\big (\bar{p},[q_1',q_2'],\epsilon r\big ) \text { for all } [q_1,q_2],[q_1',q_2']\in E_2. \end{aligned}$$

By (2), the above equality is equivalent to

$$\begin{aligned} U_1\big (\bar{p},[q_1,q_2]\big )+\bar{p}\epsilon r=&~U_1\big (\bar{p},[q_1',q_2']\big )+\bar{p}\epsilon r \text { for all } [q_1,q_2],[q_1',q_2']\in E_2. \end{aligned}$$

By (1), the claim holds if

$$\begin{aligned}&\min \left( U_1(\bar{p},q_1),U_1(\bar{p},q_2)\right) \\&\quad = ~\min \left( U_1(\bar{p},q_1'),U_1(\bar{p},q_2')\right) \text { for all } [q_1,q_2],[q_1',q_2']\in E_2. \end{aligned}$$

The remainder of the proof is a corollary of results in Riedel and Sass (2013). When \(\bar{p} \in (0,1)\), player 1 has an immunization strategy, i.e., a strategy \(p'\) such that \(U_1(p',q)=U_1(p',q')\) for all \(q,q' \in [0,1]\). Then, by Theorem 5 in Riedel and Sass (2013), the immunization strategy \(p'\) coincides with the maxmin strategy \(\bar{p}\).

1.3 Proof of Lemma 2

Proof

A proof of claim 1 can be found in Fudenberg and Tirole (1991). We prove claim 2: if Player 1’s payoffs are not Column Dominant in game \(\varGamma \), then \(\bar{p} \in (0,1)\) and \(U_1(\bar{p},q)=U_1(\bar{p},q')\) for all \(q,q' \in [0,1]\).

Geometrically, given the strategy q chosen by Player 2, the expected utility \(U_1^q(p):=U_1(p,q)\) defines a line in \([0,1]\times \mathbb {R}\). If we allow the domain of q to be \(\mathbb {R}\), this line is defined in \(\mathbb {R}^2\). The family of such lines \(\{U_1^q(p)\}_{q \in [0,1]}\) has the property of unique intersection.

Property 1

(Unique intersection)

Let \(\{U_1^q(p)\}_{q\in [0,1]}\) be a family of lines defined in \(\mathbb {R}^2\). The family has the property of unique intersection if there exists a point \((\tilde{p},u_1)\in \mathbb {R}^2\) at which all members of the family intersect.

This unique intersection \((\tilde{p},u_1)\) is hence such that for all \(q \in [0,1]\), the point \((\tilde{p},u_1)\in U_1^q(p)\).

We show now that family \(\{U_1^q(p)\}_{q \in [0,1]}\) has a unique intersection \((\tilde{p},u_1)\). Player 1’s expected utility can be rewritten:

$$\begin{aligned} U_1(p,q)= \pi _1^4+q\big ((\pi _1^3-\pi _1^4)+p(\pi _1^1-\pi _1^2-\pi _1^3+\pi _1^4)\big )+p(\pi _1^2-\pi _1^4). \end{aligned}$$

The value \(\tilde{p}\) at which an intersection takes place is therefore the solution of the following equation:

$$\begin{aligned} (\pi _1^3-\pi _1^4)+\tilde{p}(\pi _1^1-\pi _1^2-\pi _1^3+\pi _1^4)=0. \end{aligned}$$

As there are no weakly dominant strategies in \(\varGamma \), Player 1’s payoff are not Row Dominant and hence two cases can arise.

  • Case A: \(\pi _1^1>\pi _1^3\) and \(\pi _1^2<\pi _1^4\).

    The solution \(\tilde{p}\) of last equation belongs to (0, 1) if either \(\pi _1^1>\pi _1^2\) and \(\pi _1^3<\pi _1^4\) or \(\pi _1^1<\pi _1^2\) and \(\pi _1^3>\pi _1^4\). This means \(\tilde{p} \in (0,1)\) if Player 1’s payoff is not Column Dominant. Therefore, the factor \(\pi _1^1-\pi _1^2-\pi _1^3+\pi _1^4\) is different from zero. As a result, the solution \(\tilde{p}\) is unique.

  • Case B: \(\pi _1^1<\pi _1^3\) and \(\pi _1^2>\pi _1^4\)

    A parallel argument can be made to show \(\tilde{p}\in (0,1)\) and is unique.

We now prove for games with \(\tilde{p} \in (0,1)\) that this intersection is the maxmin strategy, that is \(\tilde{p}=\bar{p}\). By definition of indifference strategy \(q^*\), we have for all \(p \in [0,1]\) that \(U_1(p,q^*)=U_1(\tilde{p},q^*)=u_1\) and hence \(U_1^{q^*}(p)\) is flat: \(U_1(0,q^*)-U_1(1,q^*)=0\). As Player 1 has no weakly dominant strategy, the difference \(U_1(0,q)-U_1(1,q)\) is strictly monotone in q. This implies \(q^*\) is the only value for which \(U_1^{q}(p)\) is flat. We showed that \(q^* \in (0,1)\), implying there exist hence \(q'\) and \(q''\) in [0, 1] such that \(q'<q^*<q''\). By strict monotonicity of the difference \(U_1(0,q)-U_1(1,q)\), we have that among the two lines \(U_1^{q'}(p)\) and \(U_1^{q''}(p)\), one is strictly increasing and the other strictly decreasing. Therefore, as \(\tilde{p} \in (0,1)\), for any \(p \in [0,1]\) with \(p\ne \tilde{p} \) there exists \(q \in [0,1]\) with \(q\ne q^*\) such that \(U_1^{q}(p)<U_1^{q}(\tilde{p} )\). The maxmin strategy \(\bar{p}\) is hence at the intersection \(\tilde{p}\). This completes the proof as we showed that utility in \(\tilde{p}\) is independent of q. The proof of claim 3 is done using the same argument. \(\square \)

1.4 Proof of Lemma 3

Proof

We assume without loss of generality that \(p_1< p_2\). First, we show \(q^* \in \{q_1,q_2\}\). As shown in Lemma 2, for all \(\varGamma \in {{\varvec{\Gamma }}}\), we have that \(q^*\in (0,1)\) and is unique. The Ellsberg strategy \([p_1,p_2]\) is a best reply to \([q_1,q_2]\), if and only if we have for all \(p \in [p_1,p_2]\) there exists no \(p' \in [0,1]\) such that \(U_1\big (p',[q_1,q_2]\big )>U_1\big (p,[q_1,q_2]\big )\). Being ambiguity averse, Player 1 must be indifferent between all mixed strategies inside the Ellsberg strategy \([p_1,p_2]\) she plays. Formally, for all \(p,p'\in [p_1,p_2]\) we have

$$\begin{aligned} U_1\big (p',[q_1,q_2]\big )=U_1\big (p,[q_1,q_2]\big ). \end{aligned}$$

As \(U_1\big (p,[q_1,q_2]\big )=\min \big (U_1(p,q_1),U_1(p,q_2)\big )\) [Eq. (1)], we must have either

  • \(U_1(p,q_1)\) is constant (implying \(q_1=q^*\)) and \(U_1(p,q_1)\le U_1(p,q_2)\) for all \(p \in [p_1,p_2]\), or

  • \(U_1(p,q_2)\) is constant (implying \(q_2=q^*\)) and \(U_1(p,q_2)\le U_1(p,q_1)\) for all \(p \in [p_1,p_2]\).

Therefore, we have \(q^* \in \{q_1,q_2\}\).

Second, we show \(p^* \in \{p_1,p_2\}\). If \([q_1,q_2]\) is a proper Ellsberg strategy, then the reasoning above proves it. We show it holds as well if the equilibrium is quasi-proper, that is \(q_1=q_2\). From the previous reasoning, this implies \(q_1=q_2=q^*\). The Ellsberg strategy \(q^*\) is a best reply to \([p_1,p_2]\), if and only if there exists no \(q' \in [0,1]\) such that \(U_2\big (q',[p_1,p_2]\big )>U_2\big (q^*,[p_1,p_2]\big )\). Remember we have \(U_2\big (q,[p_1,p_2]\big )=\min \big (U_2(q,p_1),U_2(q,p_2)\big )\). Function \(U_2(q,p)\) is linear in q. We show that if \(p^* \notin \{p_1,p_2\}\), we have a contradiction.

  • If \(U_2(q,[p_1,p_2])\) is strictly increasing in q on [0, 1], then best reply is \(q_1=q_2=1\), and since \(q^*\ne 1\) for all \(\varGamma \in {{\varvec{\Gamma }}}\), we have a contradiction.

  • If \(U_2(q,[p_1,p_2])\) is strictly decreasing in q on [0, 1], then best reply is \(q_1=q_2=0\), and since \(q^*\ne 0\), we have another contradiction.

  • If \(U_2(q,[p_1,p_2])\) is strictly increasing in q on one portion of [0, 1] and strictly decreasing on the other, then the best reply is \(\bar{q}\). In effect, by the property of unique intersection, \(U_2(q,p_1)\) and \(U_2(q,p_2)\) must then intersect in \((\bar{q},u_2)\) and \(U_2(q,[p_1,p_2])\) has maximal value for \(q=\bar{q}\). As for all \(\varGamma \in {{\varvec{\Gamma }}}\), \(q^*\ne \bar{q}\), we have yet another contradiction.

The only possibility for \(q^*\) to belong to best replies is that either \(U_2(q,p_1)\) or \(U_2(q,p_2)\) is constant in q, which implies \(p^* \in \{p_1,p_2\}\). \(\square \)

1.5 Proof of Lemma 4

Proof

We show that for all \(p \in [p_{\min },p_{\max }]\), there exists a density \(f \in {\mathcal {P}}_r\) such that

$$\begin{aligned} p=\int _{-1}^{1}p^b(r)f(r)\text {d}r. \end{aligned}$$

The domain \({\mathcal {P}}_r\) is convex. This means that for all \(f_1,f_2 \in {\mathcal {P}}_r\), distribution \(f_3\) defined as \(f_3(r):=\lambda f_1(r) + (1-\lambda ) f_2(r)\) belongs to \({\mathcal {P}}_r\). The mapping \(\int _{-1}^{1}p^b(r)f(r)\text {d}r\) is linear in f. As the image of a convex set through a linear mapping is a convex set, the image of \({\mathcal {P}}_r\) through this mapping is convex. In the real line, a convex set is an interval. As \({\mathcal {P}}_r\) is closed, so must be its image \(\big [p_{\min },p_{\max }\big ]\). \(\square \)

1.6 Proof of Lemma 5

Take any \(\epsilon >0\), any \(\varGamma \in {{\varvec{\Gamma }}}\) such that Player 1’s payoffs are not Column Dominant and any \([q_{\min },q_{\max }]\subseteq [0,1]\). Given Eqs. (1) and (2), we have

$$\begin{aligned} U_1\big (p,[q_{\min },q_{\max }],\epsilon r\big )&=\min \big (U_1(p,q_{\min },\epsilon r),U_1(p,q_{\max },\epsilon r)\big )\\&=\min \big (U_1(p,q_{\min })+p\epsilon r,U_1(p,q_{\max })+p\epsilon r\big ) \end{aligned}$$

where \(U_1(p,q,\epsilon r)\) is linear in p since \(U_1(p,q)\) is linear in p. Let \(q^1:=q_{\min }\) and \(q^2 :=q_{\max }\) if \(r_1=r'\) and \(q^1 :=q_{\max }\) and \(q^2:=q_{\min }\) otherwise. By definition of \(r_1\) and \(r_2\), we have

$$\begin{aligned}&U_1(0,q^1)+0\epsilon r_1=U_1(1,q^1)+1\epsilon r_1 \\&\quad \text { and } ~~~ U_1(0,q^2)+0\epsilon r_2=U_1(1,q^2)+1\epsilon r_2. \end{aligned}$$

Remembering that \(r_1\le r_2\), these definitions imply

  • \(U_1(1,q^1)+1\epsilon r< U_1(0,q^1)+0\epsilon r\) for all \(r<r_1\),

  • \(U_1(1,q^2)+1\epsilon r< U_1(0,q^2)+0\epsilon r\) for all \(r<r_1\).

The last two inequalities imply that for all \(r<r_1\), both \(U_1(p,q^1,\epsilon r)\) and \(U_1(p,q^2,\epsilon r)\) are strictly decreasing in p because they both are linear in p. Therefore, the unique best reply when \(r<r_1\) is to take \(p=0\). The same definitions also imply that

  • \(U_1(1,q^1)+1\epsilon r> U_1(0,q^1)+0\epsilon r\) for all \(r>r_2\),

  • \(U_1(1,q^2)+1\epsilon r> U_1(0,q^2)+0\epsilon r\) for all \(r>r_2\).

The last two inequalities imply that for all \(r>r_2\), both \(U_1(p,q^1,\epsilon r)\) and \(U_1(p,q^2,\epsilon r)\) are strictly increasing in p. Therefore, the unique best reply when \(r>r_2\) is to take \(p=1\). Finally, we have

  • \(U_1(1,q^1)+1\epsilon r< U_1(0,q^1)+0\epsilon r\) for all r with \(r_1<r<r_2\),

  • \(U_1(1,q^2)+1\epsilon r> U_1(0,q^2)+0\epsilon r\) for all r with \(r_1<r<r_2\).

For those intermediate values of r, \(U_1(p,q^1,\epsilon r)\) is strictly decreasing in p while \(U_1(p,q^2,\epsilon r)\) is strictly increasing in p. By definition of \(\bar{p}\), \(U_1(p,q^1,\epsilon r)\) and \(U_1(p,q^2,\epsilon r)\) cross in \(p=\bar{p}\). By Lemma 2, \(\bar{p}\) is unique and belongs to (0, 1) since Player 1’s payoffs are not Column Dominant. The unique best reply is to take \(p=\bar{p}\). Finally, when \(r=r_1\) or \(r=r_2\), either \(U_1(p,q_{\min },\epsilon r)\) or \(U_1(p,q_{\max },\epsilon r)\) is constant in p. A (non-unique) best reply is then \(p=\bar{p}\). Notice that this proof also covers the case \(q_{\min }=q_{\max }\).

1.7 Proof of Lemma 6

Take any \(\epsilon >0\), any \(\varGamma \in {{\varvec{\Gamma }}}^{IV}\), any \((k_r,k_t) \in [0,1]\times [0,1]\) and any profile of pure–or–maxmin strategies \(\big ((r_1,r_2),(t_1,t_2)\big ) \in S^1_{np}\times S^2_{np}\) for which Eqs. (10) to (17) hold. From Lemma 4, the endpoints of the induced Ellsberg strategies \([p_{\min },p_{\max }]\) and \([q_{\min },q_{\max }]\) of any profile \(\big ((r_1,r_2),(t_1,t_2)\big )\) are given by Eqs. (14) to (17). As \(\varGamma \in {{\varvec{\Gamma }}}^{IV}\), we have that Player 1’s payoffs are not Column Dominant. From Lemma 5, the best reply of Player 1 to \([q_{\min },q_{\max }]\) is to use a strategy \((r_1,r_2)\) whose thresholds \(r_1\) and \(r_2\) are defined by Eqs. (10) and (12). Accordingly, the best reply for Player 2 to \([p_{\min },p_{\max }]\) is a pure–or–maxmin strategy \((t_1,t_2)\), whose thresholds \(t_1\) and \(t_2\) are defined by Eqs. (11) and (13). Therefore, if all equations hold, strategies \((r_1,r_2)\) and \((t_1,t_2)\) are mutual best replies and the profile constitutes an equilibrium in \(\varGamma ^*(\epsilon )\).

1.8 Proof of Lemma 7

Take any \(\epsilon >0\) and any \(\varGamma \in {{\varvec{\Gamma }}}\). We focus on proving this for the interval \(\big (p^-(\epsilon ),p^+(\epsilon )\big )\), the reasoning is identical for \(\big (q^+(\epsilon ),q^-(\epsilon )\big )\). The expression

$$\begin{aligned} \frac{1}{\epsilon }\big (U_2(0,p)-U_2(1,p)\big ) \end{aligned}$$

returning the thresholds on t is linear in p. As \(\varGamma \in {{\varvec{\Gamma }}}\), Player 2 does not have a weakly dominant strategy, and therefore, this expression is strictly monotone in p. Therefore, \(p^-(\epsilon )\) and \(p^+(\epsilon )\) are finite and hence exist. By definition of \(p^*\), this expression equals 0 for \(p=p^*\). Therefore, we have \(p^-(\epsilon )< p^*< p^+(\epsilon )\) by the strict monotonicity of the above linear expression.

The difference \(U_2(0,p)-U_2(1,p)\) is independent of \(\epsilon \). As a result, for any \(p\ne p^*\) the smaller \(\epsilon \), the larger \(\big \arrowvert \frac{1}{\epsilon }\big (U_2(0,p)-U_2(1,p)\big ) \big \arrowvert \). Hence, for any \(p\ne p^*\), there exists an \(\epsilon ^p\) such that for all \(\epsilon <\epsilon ^p\), we have \(\frac{1}{\epsilon }\big (U_2(0,p)-U_2(1,p)\big ) \notin [-1,1]\). Therefore, the smaller \(\epsilon \), the closer \(p^-(\epsilon )\) and \(p^+(\epsilon )\) are to \(p^*\). In the limit, the interval \([p^-(\epsilon ),p^+(\epsilon )]\) collapse on \(p^*\).

1.9 Two lemmata for small perturbations

Lemmata 12 and 13 provide bounds around the endpoints of the induced Ellsberg strategies when thresholds lie in the interior of the support and ambiguity is strictly positive. If the domain \({\mathcal {P}}_r\) contains a strictly positive amount of ambiguity, Lemma 12 shows that the induced Ellsberg strategy cannot degenerate into a mixed strategy when a threshold lies in the interior of the support. If one endpoint of the induced Ellsberg strategy for Player 1 equals p, the other endpoint lies outside a non-degenerate interval \((p^l,p^u)\) around p. New notation is necessary for establishing Lemma 12 besides the notations A, B, C, and D defined just before Lemma 8.

  • Let function \(p_{\min }:\mathbb {R}^2\rightarrow [0,1]:(r_1,r_2)\rightarrow p_{\min }(r_1,r_2)\) be defined by Eq. (14). Accordingly, functions \(p_{\max }\), \(q_{\min }\) and \(q_{\max }\) are defined by Eqs. (15)–(17), respectively.

Lemma 12

For all \(\varGamma \in {{\varvec{\Gamma }}}^{IV}\), \(k_r \in (0,1]\) and \(p \in (A,B)\), there exist unique \(p^l\) and \(p^u \in [0,1]\) with \(p^l<p<p^u\) such that

  • for all \((r_1,r_2) \in S^1_{np}\) with \(p_{\max }(r_1,r_2)=p\), we have \(p_{\min }(r_1,r_2)\le p^l\);

    and at least for one such \((r_1,r_2)\) we have \(p_{\min }(r_1,r_2)= p^l\),

  • for all \((r_1,r_2) \in S^1_{np}\) with \(p_{\min }(r_1,r_2)=p\), \(p_{\max }(r_1,r_2)\ge p^u\),

    and at least for one such \((r_1,r_2)\) we have \(p_{\max }(r_1,r_2)= p^u\),

Accordingly, for all \(k_t \in (0,1]\) and \(q \in (C,D)\), there exist \(q^l,q^u \in [0,1]\) with equivalent properties.

Proof

Take any \(\varGamma \in {{\varvec{\Gamma }}}^{IV}\), any \(k_r \in (0,1]\) and any \(p \in (A,B)\). We focus on proving the existence of such \(p^l\) and \(p^u\). The proof for \(q^l\) and \(q^u\) follows the same reasoning. We define the following sets:

$$\begin{aligned}&S^{\max }(p):=\big \{(r_1,r_2)\in S^1_{np} \big \arrowvert p_{\max }(r_1,r_2)=p~ \text { and }~ r_1,r_2 \in [-1,1] \big \},\\&S^{\min }(p):=\big \{(r_1,r_2)\in S^1_{np} \big \arrowvert p_{\min }(r_1,r_2)=p~\text { and }~ r_1,r_2 \in [-1,1]\big \}. \end{aligned}$$

\(S^{\max }(p)\) is a subset of the pure–or–maxmin strategies whose induced Ellsberg strategies have p as their maximal point. We show below this set is non-empty. The restriction \(r_1,r_2 \in [-1,1]\) implies that \(S^{\max }(p)\) and \(S^{\min }(p)\) are closed sets.

We define \(p^l\) from the set \(S^{\max }(p)\):

$$\begin{aligned} p^l:=\max _{(r_1,r_2) \in S^{\max }(p)} p_{\min }(r_1,r_2). \end{aligned}$$

As the domain of images of function \(p_{\min }\) is [0, 1] and the set \(S^{\max }(p)\) is non-empty and closed, \(p^l\) is well defined. The definitions of \(p^l\) and \(S^{\max }(p)\) imply that \(p^l\) is such that:

  1. (i)

    for all \((r_1,r_2) \in S^{\max }(p)\) we have \(p_{\max }(r_1,r_2)=p\) and \(p_{\min }(r_1,r_2)\le p^l\); at least for one such \((r_1,r_2)\) we have \(p_{\min }(r_1,r_2)= p^l\), and

  2. (ii)

    there is no \(p'\ne p^l\) with the previous properties.

We next show that \(p^l<p\). As \(\varGamma \in {{\varvec{\Gamma }}}^{IV}\) we have \(\bar{p}\in (A,B)\), and hence, two cases can arise:

  • Case 1: \(p^*>\bar{p}\). This case is such that \(A=\bar{p}\) and \(B=1\) and by assumption we have \(p\in (\bar{p},1)\). Let \(r_2^L\) and \(r_2^H\) be implicitly defined by

    $$\begin{aligned} p_{\max }\left( -1,r_2^H\right) =p \qquad \text { and } \qquad p_{\max }\left( r_2^L,r_2^L\right) =p. \end{aligned}$$

    We show that for all \((r_1,r_2) \in S^{\max }(p)\), we have \(-1<r_2^L\le r_2 \le r_2^H<1\). Observe this implies that \(S^{\max }(p)\) is a non-empty set.

    • First we show \(-1<r_2^L<1\).

      For all \(k_r \in (0,1]\), because of its integral functional form, the expression of \(p_{\max }(x,x)\) is continuous in x. Furthermore, it is decreasing in x for \(x \in [-1,1)\) as pure–or–maxmin strategies are increasing in r. Since \(p_{\max }(-1,-1)=1\), \(p_{\max }(1,1)=0\) and by assumption \(p\in (\bar{p},1)\), we therefore have \(-1<r_2^L<1\).

    • Second we show \(-1<r_2^H<1\).

      For all \(k_r \in (0,1]\), the expression of \(p_{\max }(-1,x)\) is continuous in x and decreasing in x for \(x \in [-1,1)\). Since \(p_{\max }(-1,-1)=1\), \(p_{\max }(-1,1)=\bar{p}\) and by assumption \(p\in (\bar{p},1)\), we therefore have \(-1<r_2^H<1\).

    • Then we show that \(r_2^L<r_2^H\).

      Assume instead that \(r_2^L\ge r_2^H\). As by definition \(p_{\max }\left( -1,r_2^H\right) =p\), we have \(p_{\max }\left( r_2^H,r_2^H\right) <p\) as for all \(k_r \in (0,1]\) and \(r_1,r_2 \in [-1,1)\), \(p_{\max }\) is a strictly decreasing function of both \(r_1\) and \(r_2\) and we showed \(-1<r_2^H\). As \(r_2^L\ge r_2^H\) the same reasoning implies \(p_{\max }\left( r_2^L,r_2^L\right) \le p_{\max }\left( r_2^H,r_2^H\right) <p\), contradiction the definition of \(r_2^L\).

    • Finally we show that for all \((r_1,r_2) \in S^{\max }(p)\) we have \(r_2^L\le r_2 \le r_2^H\).

      We focus on showing \(r_2 \le r_2^H\), the proof that \(r_2^L\le r_2\) follows similar lines. Assume instead for some \((r_1,r_2) \in S^{\max }(p)\) that \(r_2 > r_2^H\). By the definition of \(S^{\max }(p)\) we have \(-1\le r_1\). As \(p_{\max }\) is strictly decreasing in its argument, this implies that \(p_{\max }(r_1,r_2)\le p_{\max }(-1,r_2)\). As we assumed \(r_2 > r_2^H\), the same reasoning implies \(p_{\max }(-1,r_2)< p_{\max }(-1,r_2^H)=p\). Together we have \(p_{\max }(r_1,r_2)<p\), implying that \((r_1,r_2) \notin S^{\max }(p)\), a contradiction.

  • Case 2: \(p^*<\bar{p}\). This second case is such that \(A=0\) and \(B=\bar{p}\) and by assumption we have \(p\in (0,\bar{p})\). Let \(r_1^L\) and \(r_1^H\) be implicitly defined by \(p_{\max }(r_1^H,1)=p\) and \(p_{\max }(r_1^L,r_1^L)=p\). The proof showing that for all \((r_1,r_2) \in S^{\max }(p)\), we have \(-1<r_1^L\le r_1 \le r_1^H<1\) is omitted as it follows the lines of that given for case 1.

Together, either there exist \(r_1^L\) and \(r_1^H\) such that for all \((r_1,r_2) \in S^{\max }(p)\) we have \(-1<r_1^L\le r_1 \le r_1^H<1\) or there exist \(r_2^L\) and \(r_2^H\) such that for all \((r_1,r_2) \in S^{\max }(p)\) we have \(-1<r_2^L\le r_2 \le r_2^H<1\). This implies \(min\{\mid r_1\mid ,\mid r_2\mid \}<1\).

From there, as \(k_r>0\) we have for all \((r_1,r_2) \in S^{\max }(p)\) that

$$\begin{aligned} p_{\min }(r_1,r_2)<p_{\max }(r_1,r_2) \end{aligned}$$

because

  1. (i)

    \(p_{\min }(r_1,r_2)=p_{\max }(r_1,r_2)\) when \(k_r=0\) and,

  2. (ii)

    for all \((r_1,r_2)\) with \(min\{\mid r_1\mid ,\mid r_2\mid \}<1\), \(p_{\max }\) is a strictly increasing function of \(k_r\) at all \(k_r \in [0,1)\), while \(p_{\min }\) is a strictly decreasing function of \(k_r\).

This proves that \(p^l<p\).

There remains to show that \(p^l\) has the same properties for all \((r_1,r_2)\in S^1_{np}\). As the support of r is \([-1,1]\), for any \((r_1,r_2)\in S^1_{np}\) with \(p_{\max }(r_1,r_2)=p\) such that \((r_1,r_2)\notin S^{\max }(p)\), there exists \((r_1',r_2') \in S^{\max }(p)\) inducing the same Ellsberg strategy as \((r_1,r_2)\). Therefore \(p^l\) has the desired properties.

We define then \(p^u\) from the set \(S^{\min }(p)\):

$$\begin{aligned} p^u:=\min _{(r_1,r_2) \in S^{\min }(p)} p_{\max }(r_1,r_2). \end{aligned}$$

An analog reasoning proves that \(p^u\) has the desired properties. \(\square \)

Lemma 13 shows that the interval \((p^l,p^u)\) around p defined in the previous lemma evolves monotonically with p.

Lemma 13

Take any \(\varGamma \in {{\varvec{\Gamma }}}^{IV}\).

  • For all \(k_r \in (0,1)\), \(p \in (A,B)\) and \(p'\in ({p}^l,p)\), we have

    $$\begin{aligned} {p'}^l<p^l<p'<p<{p'}^u<p^u. \end{aligned}$$
  • For all \(k_t \in (0,1)\), \(q \in (C,D)\) and \(q'\in ({q}^l,q)\), we have

    $$\begin{aligned} {q'}^l<q^l<q'<q<{q'}^u<q^u. \end{aligned}$$

Proof

We focus on proving the first claim. The proof is based on the properties of functions \(p_{\min }\) and \(p_{\max }\). Those functions are continuous in both their arguments \(r_1\) and \(r_2\). Furthermore, they are non-increasing in both arguments and strictly decreasing as soon as these arguments belong to \([-1,1)\).

Take any \(\varGamma \in {{\varvec{\Gamma }}}^{IV}\), any \(k_r \in (0,1)\), any \(p \in (A,B)\) and any \(p'\in ({p}^l,p)\). From Lemma 12, we have that \({p'}^l<p^l<{p'}^u\). We show by contradiction that \({p'}^u< p^u\), \( p <{p'}^u\) and \({p'}^l<{p}^l\).

Assume first \( p^u \le {p'}^u\). This implies by definition of \({p'}^u\) that there does not exist \((r_1,r_2)\) with \(p_{\min }(r_1,r_2)=p'\) and \(p_{\max }(r_1,r_2)<p^u\). Take \((r_1',r_2')\) with \(p_{\min }(r_1',r_2')=p\) and \(p_{\max }(r_1',r_2')=p^u\). By definition of \(p^u\), this \((r_1',r_2')\) exists and has at least one threshold in the interior of the support. By continuity and non-increasingness of \(p_{\min }\), there exists \((r_1,r_2)\) with \(r_1>r_1'\) and \(r_2>r_2'\) such that \(p_{\min }(r_1,r_2)=p'\). Since \(p \in (A,B)\), we have either \(r_1'\in (-1,1)\) or \(r_2'\in (-1,1)\).Footnote 12 By the properties of \(p_{\max }\), we have \(p_{\max }(r_1,r_2)<p_{\max }(r_1',r_2')=p^u\), a contradiction.

Assume then that \({p'}^u\le p\). This implies by definition of \({p'}^u\) that there exists \((r_1,r_2)\) with \(p_{\min }(r_1,r_2)=p'\) and \(p_{\max }(r_1,r_2)\le p\). By continuity and non-increasingness of \(p_{\max }\), there exists \((r_1',r_2')\) with \(r_1'\le r_1\) and \(r_2'\le r_2\) such that \(p_{\max }(r_1',r_2')=p\). By the properties of \(p_{\min }\), we have \(p_{\min }(r_1',r_2')\ge p_{\min }(r_1,r_2)=p'\), a contradiction to the definition of \(p^l\) since \(p^l<p'\).

Assume finally that \({p}^l\le {p'}^l\). This implies by definition of \({p'}^l\) that there exists \((r_1,r_2)\) with \(p_{\max }(r_1,r_2)=p'\) and \(p_{\min }(r_1,r_2)\ge p^l\). By continuity and non-increasingness of \(p_{\max }\), there exists \((r_1',r_2')\) with \(r_1'< r_1\) and \(r_2'< r_2\) such that \(p_{\max }(r_1',r_2')=p\). By the properties of \(p_{\min }\), we have \(p_{\min }(r_1',r_2')> p_{\min }(r_1,r_2)\ge p^l\), which contradicts the definition of \({p}^l\). \(\square \)

1.10 Proof of Lemma 8

We show that such pure–or–maxmin strategies \((r_1,r_2)\) and \((t_1,t_2)\) are mutual best replies. Take any \(\varGamma \in {{\varvec{\Gamma }}}^{IV4}\). Lemma 6 gives sufficient conditions for such profile to be an equilibrium. In these conditions, the following additional four equations complement Eqs. (22) to (25):

$$\begin{aligned} \epsilon r'&= U_1(0,q_{\min })-U_1(1,q_{\min }),\\ \epsilon t'&= U_2(0,p_{\min })-U_2(1,p_{\min }),\\ p_{\min }&=\min _{f\in {\mathcal {P}}_r} \left( \int _{-1}^{r^*}Af(r)\text {d}r+\int _{r^*}^{1}Bf(r)\text {d}r\right) ,\\ q_{\min }&=\min _{f\in {\mathcal {P}}_t} \left( \int _{-1}^{t^*}Cf(t)\text {d}t+\int _{t^*}^{1}Df(t)\text {d}t\right) . \end{aligned}$$

We show there exists \(\overline{\epsilon }>0\) such that for all \(\epsilon <\overline{\epsilon }\), if \(\big ((r_1,r_2),(t_1,t_2)\big )\) satisfy conditions (18) to (21), then \(r' \notin [-1,1]\) and \(t' \notin [-1,1]\), and hence, those four additional equations are irrelevant for the profile to be an equilibrium.

By definition of A and B, we have \(p^* \in (A,B)\). By Lemma 7, there exists \(\overline{\epsilon }_1>0\) such that for all \(\epsilon <\overline{\epsilon }_1\) we have \(\big [p^-(\epsilon ),p^+(\epsilon )\big ] \subset (A,B)\). Accordingly, we have \(q^* \in (C,D)\) and there exists \(\overline{\epsilon }_2>0\) such that for all \(\epsilon <\overline{\epsilon }_2\) we have \(\big [q^-(\epsilon ),q^+(\epsilon )\big ] \subset (C,D)\). As \(\big ((r_1,r_2),(t_1,t_2)\big )\) satisfies conditions (18) to (21), we have \(r^*\in [-1,1]\) and \(t^*\in [-1,1]\), and hence, two cases must be considered.

Case 1: \(\arrowvert r^*\arrowvert =1\) or \(\arrowvert t^*\arrowvert =1\).

Taking \(\overline{\epsilon }=min\{\overline{\epsilon }_1,\overline{\epsilon }_2\}\) we derive a contradiction for this case. Assume that \(\arrowvert t^*\arrowvert =1\). Conditions (20) and (21) imply that \(min\{\mid t_1\mid ,\mid t_2\mid \}\ge 1\). The pure–or–maxmin strategy \((t_1,t_2)\) is such that \(q^b(t)\) is the same for all \(t \in [-1,1]\) with \(q^b(t)\in \{C,D\}\) and therefore \(q_{\min }=q_{\max } \in \{C,D\}\). As for all \(\epsilon <\overline{\epsilon }\), we have \(\big [q^-(\epsilon ),q^+(\epsilon )\big ] \subset (C,D)\), this implies that either \(r^* \notin [-1,1]\), which violates condition (18) or (19), or Eq. (22) does not hold.

Case 2: \(\arrowvert r^*\arrowvert <1\) or \(\arrowvert t^*\arrowvert <1\).

Proving that \((r_1,r_2)\) and \((t_1,t_2)\) are mutual best replies boils down to showing that

  1. (i)

    \(r'\) and \(t'\) are not in the support and,

  2. (ii)

    the relative size of \(r^*\) and \(r'\) makes it optimal for Player 1 to react to r using strategies A and B, as well as it is optimal for Player 2 to react to t using C and D given the relative size of \(t^*\) and \(t'\).

If (i) and (ii) hold, then Eqs. (22) to (25) are a simplification of Eqs. (10) to (17) and the strategies are mutual best replies. Two subcases must be considered

  • Subcase 2.1: \(k_r>0\) and \(k_t>0\).

    The profile of pure–or–maxmin strategies \(\big ((r_1,r_2),(t_1,t_2)\big )\) induces proper Ellsberg strategies since thresholds \(r^*\) and \(t^*\) lie in the interior of \([-1,1]\).Footnote 13 Player 1’s proper Ellsberg strategy has two different endpoints \(p_{\max }\) and \(p_{\min }\) which induce two different thresholds \(t^*\) and \(t'\) for Player 2. Accordingly, we have \(q_{\max }\ne p_{\min }\) and hence \(t^*\ne t'\).

    We show here (i) that is \(r'\) and \(t'\) are not in the support. Given \(k_r>0\) and \(k_t>0\), by Lemmas 12 and  13 there exist \(p_L\), \(p_U \in (A,B)\) and \(q_L\), \(q_U \in (C,D)\) such that

    $$\begin{aligned}&p_L^l\le p_U^l<p_L<p^*<p_U<p_L^u\le p_U^u,\\&q_L^l\le q_U^l<q_L<q^*<q_U<q_L^u\le q_U^u. \end{aligned}$$

    such that if \(p_{\max } \in [p_L,p_U]\), then \(p_{\min } \notin [p_L,p_U]\) and if \(q_{\max } \in [q_L,q_U]\), then \(q_{\min } \notin [q_L,q_U]\). We prove the existence of such \(p_L\) and \(p_U\). As \(p^* \in (A,B)\), given \(k_r>0\), Lemma 12 shows there exists \({p^*}^l\) and \({p^*}^u\) with \({p^*}^l<p^*<{p^*}^u\) such that if \(p_{\max } =p^*\), then \(p_{\min } \le {p^*}^l\) and if \(p_{\min } =p^*\), then \(p_{\max } \ge {p^*}^u\). Take any \(p_L \in ({p^*}^l,p^*)\). By Lemma 13, we have \(p^*<p_L^u\). Take \(p_U\) such that \(p^*<p_U<p_L^u\). By Lemma 13, we have \(p_L^l<p_U^l<p_L<p^*<p_U<p_L^u<p_U^u\) and hence the desired property for \([p_L,p_U]\).Footnote 14

    Let \(\overline{\epsilon }'>0\) be such that \(\overline{\epsilon }'\le \overline{\epsilon }_1\) and for all \(\epsilon <\overline{\epsilon }'\) we have \(\big [p^-(\epsilon ),p^+(\epsilon )\big ] \subset (p_L,p_U)\). By Lemma 7, this \(\overline{\epsilon }'\) exists since \(\big [p^-(\epsilon ),p^+(\epsilon )\big ]\) tends to \([p^*,p^*]\) as \(\epsilon \rightarrow 0\). The same reasoning proves the existence of an \(\overline{\epsilon }''>0\) such that \(\overline{\epsilon }''\le \overline{\epsilon }_2\) and for all \(\epsilon <\overline{\epsilon }''\) we have \(\big [q^-(\epsilon ),q^+(\epsilon )\big ] \subset (q_L,q_U)\). Take \(\overline{\epsilon }=min\{\overline{\epsilon }',\overline{\epsilon }''\}\).

    By the construction of \(\overline{\epsilon }\), for all \(\epsilon <\overline{\epsilon }\) conditions (18) and (19) combined with Eq. (22) imply that \(q_{\max } \in \big [q^-(\epsilon ),q^+(\epsilon )\big ] \subset [q_L,q_U]\) and hence \(q_{\min } \notin [q_L,q_U]\), therefore \(q_{\min } \notin \big [q^-(\epsilon ),q^+(\epsilon )\big ]\), implying \(r' \notin [-1,1]\). A parallel reasoning shows \(t' \notin [-1,1]\).

    We turn to proving (ii). We focus on showing that the relative sizes of \(t^*\) and \(t'\) make it optimal for Player 2 to react to t using strategies C and D. A parallel argument demonstrates that Player 1 best replies using A and B. As \(\varGamma \in {{\varvec{\Gamma }}}^{D - 4}\), we have \(\bar{p}\in (A,B)\), and hence, two subcases can arise:

    • Subcase 2.1.1: \(\bar{q}<q^*\).

      Assume for a moment that the difference \(U_2(0,p)-U_2(1,p)\) is a strictly increasing function of p. As \(p_{\min }<p_{\max }\), this assumption implies that for a given \(\epsilon \) we have \(t'<t^*\) and hence \(t_1=t'\) and \(t_2=t^*\). As \(t^* \in [-1,1]\) and \(t' \notin [-1,1]\), we have \(t_1<-1\). By definition, when \(\bar{q}<q^*\), we have \(C=\bar{q}\) and \(D=1\). It is hence optimal for Player 2 to react to the realization of t using C and D, as shown in the proof of Lemma 5.

      There remains to show that the difference \(U_2(0,p)-U_2(1,p)\) is a strictly increasing function of p. The difference \(U_2(0,p)-U_2(1,p)\) is linear in p and cannot be constant since weakly dominant strategies are ruled out. By definition, any game \(\varGamma \in {{\varvec{\Gamma }}}^{IV4}\) has proper Ellsberg equilibria \(e=\big ([p_1,p_2],[q_1,q_2]\big )\) of type 4, for which \(p_2=p^*\) and hence \(p_1<p^*\). In order for \([q_1,q_2]\) to be a best reply to \([p_1,p_2]\), we must have \(q^*\in \{q_1,q_2\}\) as shown in Lemma 3. For \(q^*\in \{q_1,q_2\}\), we must have

      $$\begin{aligned} U_2(q,p_1)> U_2(q,p^*) \text { for all } q \in (\bar{q},1]. \end{aligned}$$

      In effect, remember that the definition of \(p^*\) implies that \(U_2(0,p^*)-U_2(1,p^*)=0\) and hence \(U_2(q,p^*)\) is constant in q. The definition of \(\bar{q}\) implies \(U_2(\bar{q},p^*)=U_2(\bar{q},p_1)\). If we had instead for all \(q \in (\bar{q},1]\) that \(U_2(q,p_1)< U_2(q,p^*)\), then \(U_2(q,p_1)\) is strictly decreasing in q and the best reply for the ambiguity-averse Player 2 to \([p_1,p^*]\) would be some \([q_1,q_2] \subset [0,\bar{q}]\), contradicting Lemma 3 since \(\bar{q}<q^*\). As \(U_2(\bar{q},p^*)=U_2(\bar{q},p_1)\) and \(U_2(1,p_1)> U_2(1,p^*)\), we have \(U_2(0,p_1)< U_2(0,p^*)\). Last two inequalities imply \(U_2(0,p_1)-U_2(1,p_1)<U_2(0,p^*)-U_2(1,p^*)\) and by linearity of \(U_2(q,p)\) in p, the difference \(U_2(0,p)-U_2(1,p)\) is a strictly increasing function of p as \(p_1<p^*\).

    • Subcase 2.1.2: \(q^*<\bar{q}\).

      The argument follows the same line as for the previous case. The major difference is that \(U_2(0,p)-U_2(1,p)\) must now be strictly decreasing function of p. As \(p_{\min }<p_{\max }\), this implies that for a given \(\epsilon \) we have \(t^*<t'\), and hence, \(t_1=t^*\) and \(t_2=t'\). As \(t^* \in [-1,1]\) and \(t' \notin [-1,1]\), we have \(t_2>1\). By definition, when \(\bar{q}<q^*\), we have \(C=0\) and \(D=\bar{q}\). It is hence optimal for Player 2 to react to the realization of t using C and D.

    Statement (ii) holds as for each of the above subcases, conditions (20) and (21) pick \(t^*\) among \(t_1\) and \(t_2\) consistently with the particular game considered and ensure that \(t'\) is outside the support with the appropriate relative size with respect to \(t^*\).

  • Subcase 2.1: \(k_r=0\) or \(k_t=0\).

    We consider only \(k_r=0\), without loss of generality. This implies that \(C=0\) and \(D=1\), \(p_{\min }=p_{\max }\) and \(t'=t^*\). Both \(t'\) and \(t^*\) belong to the interior of the support. The induced Ellsberg profile is quasi-proper. Except for these differences, the argument given above to prove (i) and (ii) carries on to this subcase.

1.11 Proof of Lemma 9

Using the Intermediate Value Theorem, we show the existence of a profile of thresholds \((r^*,t^*)\) satisfying Eqs. (22) to (25). If it exists, then it is easy to see that there always exists a strategy profile \(\big ((r_1,r_2),(t_1,t_2)\big )\) that, together with \((r^*,t^*)\), satisfies conditions (18) to (21). Such strategy profile \(\big ((r_1,r_2),(t_1,t_2)\big )\) satisfies the conditions of Lemma 8 for small \(\epsilon \). This proves the existence of equilibria in slightly perturbed games.

There remains to show the existence of a profile of thresholds \((r^*,t^*)\) satisfying Eqs. (22) to (25). Take any \(\varGamma \in {{\varvec{\Gamma }}}^{IV4}\) and any \((k_r,k_t) \in [0,1]\times [0,1]\). We define the four functions \(h_r\), \(h_t\), \(h_p\) and \(h_q\):

$$\begin{aligned}&h_r:[q^-(\epsilon ),q^+(\epsilon )]\rightarrow [-1,1]:q \rightarrow h_r(q):=\frac{1}{\epsilon }\big (U_1(0,q)-U_1(1,q)\big ),\\&h_t:[p^-(\epsilon ),p^+(\epsilon )]\rightarrow [-1,1]:p \rightarrow h_t(p):=\frac{1}{\epsilon }\big (U_2(0,p)-U_2(1,p)\big ),\\&h_p:[-1,1]\rightarrow [A,B]:r \rightarrow h_p(r):=\max _{f\in {\mathcal {P}}_r} \left( \int _{-1}^{r}Af(r)\text {d}r+\int _{r}^{1}Bf(r)\text {d}r\right) ,\\&h_q:[-1,1]\rightarrow [C,D]:t \rightarrow h_q(t):=\max _{f\in {\mathcal {P}}_t} \left( \int _{-1}^{t}Cf(t)\text {d}t+\int _{t}^{1}Df(t)\text {d}t\right) . \end{aligned}$$

Those four functions are all strictly monotone and continuous. By the definition of \(p^-(\epsilon )\), \(p^+(\epsilon )\), \(q^-(\epsilon )\) and \(q^+(\epsilon )\), the domain of images of \(h_r\) and \(h_t\) is \([-1,1]\), and hence, all four functions are surjective. The strict monotonicity of these functions implies they are injective. Being all bijective (surjective and injective), they admit inverse functions \(h_r^{-1}\), \(h_t^{-1}\), \(h_p^{-1}\), \(h_q^{-1}\) which are strictly monotone and continuous.Footnote 15

Based on these four functions, we define two composite functions \(g_1\) and \(g_2\):

$$\begin{aligned}&g_1:[A,B]\rightarrow [q^-(\epsilon ),q^+(\epsilon )]:p \rightarrow g_1(p):=h_r^{-1}\circ h_p^{-1}(p)= (h_p\circ h_r)^{-1}(p),\\&g_2:[C,D]\rightarrow [p^-(\epsilon ),p^+(\epsilon )]:q \rightarrow g_2(q):=h_t^{-1}\circ h_q^{-1}(q)= (h_q\circ h_t)^{-1}(q). \end{aligned}$$

Being composite functions of strictly monotone and continuous functions, \(g_1\) and \(g_2\) inherit those properties.

By Lemma 7, there exists \(\overline{\epsilon }'>0\) such that for all \(\epsilon <\overline{\epsilon }'\) we have \([p^-(\epsilon ),p^+(\epsilon )] \subset (A,B)\) and \([q^-(\epsilon ),q^+(\epsilon )] \subset (C,D)\). Those two composite functions are then used to define the continuous mapping \(\tau \):

$$\begin{aligned} \tau :[A,B]\rightarrow [p^-(\epsilon ),p^+(\epsilon )]:p \rightarrow \tau (p):=g_2\circ g_1(p). \end{aligned}$$

We have therefore that for all \(\epsilon <\overline{\epsilon }'\), \(\tau \) is a continuous mapping from \([A,B]\rightarrow [p^-(\epsilon ),p^+(\epsilon )] \subset (A,B)\). By the Intermediary Value Theorem, it has a fixed point \(\hat{p}\in [p^-(\epsilon ),p^+(\epsilon )]\). This fixed point is associated with \(\hat{q}=g_1(\hat{p})\) as well as \(\hat{r}=h_r(\hat{q})\) and \(\hat{t}=h_t(\hat{p})\). By construction, these \(\hat{r}\), \(\hat{t}\), \(\hat{p}\) and \(\hat{q}\) satisfy Eqs. (22) to (25) in Lemma 8. Let \(\overline{\epsilon }''\) be taken from the statement of Lemma 8. Taking \(\overline{\epsilon }=min\{\overline{\epsilon }',\overline{\epsilon }''\}\) completes the proof.

1.12 Proof of Lemma 10

We prove only the existence and uniqueness of \(r^*\). Function \(p_{\min }\) is continuous and weakly decreasing in \(r^*\).Footnote 16 Furthermore, \(p_{\min }(-1)=B\) and \(p_{\min }(1)=A\). By continuity, there exists hence \(r^* \in (-1,1)\) such that \(p_{\min }(r^*)=p\). We prove now uniqueness.

For all \(k_r \in [0,1)\), all \(f \in {\mathcal {P}}_r\) have full support. Function \(p_{\min }\) is therefore strictly decreasing for all \(r \in [-1,1]\), which entails uniqueness of \(r^*\).

For the case \(k_r=1\), let \(R_A:=\{r\in [-1,1]\arrowvert p_{\min }(r)=A\}\) and let \(\hat{r} :=\min \{r \in R_A\}\). This \(\hat{r}\) exists since the set \(R_A\) is a non-degenerate closed interval. Uniqueness of \(r^*\) is ensured since \(p>A\) and for all \(r \in [-1,\hat{r})\), function \(p_{\min }\) is strictly decreasing in r.

1.13 Proof of Lemma 11

We prove only the first of the two claims.

By Lemma 10, for all \(p_1 \in (A,B)\) and all \(k_r \in [0,1]\), there exists a unique \(r^*\in (-1,1)\) such that \(p_{\min }(r^*)=p_1\). Let \(F:[0,1]\rightarrow [-1,1]:k_r\rightarrow F(k_r)\) be the function pointing, for each value of \(k_r \in [0,1]\), to the particular \(r^*\) inducing \(p_{\min }(r^*)=p_1\) and hence \(F(k_r)=r^*\). From Eq. (26), function F is continuous and strictly decreasing in \(k_r\) as \(p_1 \in (A,B)\).

As F is continuous, the composite function \(p_{\max }\circ F:[0,1]\rightarrow [p_1,b]: k_r \rightarrow p_{\max }\big (F(k_r)\big )\) is continuous and strictly increasing in \(k_r\) as \(p_1 \in (A,B)\). The first claim we need to prove follows then from the fact that \(p_{\max }\big (F(0)\big )=p_{\min }\big (F(0)\big )=p_1\) and \(p_{\max }\big (F(1)\big )=B\).

The equality \(p_{\max }\big (F(1)\big )=B\) follows from the definition of the domain \({\mathcal {P}}_r\). For \(k_r=1\), some \(f \in {\mathcal {P}}_r\) do not have full support anymore, and there exists a unique \(\hat{r} \in {(-1,1)}\) such that:

$$\begin{aligned} \max _{f \in {\mathcal {P}}_r}\int _{-1}^{\hat{r}}f(r)\text {d}r=1 \qquad \text {and} \qquad \min _{f \in {\mathcal {P}}_r}\int _{-1}^{\hat{r}}f(r)\text {d}r=0. \end{aligned}$$

This \(\hat{r}\) is implicitly defined by \(\int _{-1}^{\hat{r}}f^b_r(r)\text {d}r=\frac{1}{2}\). As a result, if \(p_{\min }=p_1<1\), then \(r^*<\hat{r}\) and \(p_{\max }=B\). In effect, when \(k_r=1\), for all \(r \in (-1,1)\) either \(p_{\min }(r)=A\) or \(p_{\max }(r)=B\).

1.14 Best replies in perturbed game

For all \(\varGamma \in {{\varvec{\Gamma }}}\) such that Player 1’s payoffs are Column Dominant, the following lemma provides conditions under which a pure strategy of Player 1 is a best reply to a strategy of Player 2 inducing \([q_{\min },q_{\max }]\).Footnote 17

Lemma 14

(Best reply in pure strategy)

For all \(\epsilon >0\) and all \(\varGamma \in {{\varvec{\Gamma }}}\) such that Player 1’s payoffs are Column Dominant, the strategy \(p^b\) is a best reply to any \([q_{\min },q_{\max }] \subseteq [0,1]\) if it is a pure strategy \(p^{b}=r^*\in S^1_{pu}\) defined by:

$$\begin{aligned} \epsilon r'&= U_1(0,q_{\min })-U_1(1,q_{\min }),\\ \epsilon r''&= U_1(0,q_{\max })-U_1(1,q_{\max }),\\ r^*&= \left\{ \begin{array}{rl} &{} \max \{r',r''\} \qquad \text { if }\, |\pi _1^3-\pi _1^4|\le |\pi _1^1-\pi _1^2|, \\ &{} \min \{r',r''\} \qquad \text { otherwise. } \end{array} \right. \end{aligned}$$

Proof

The proof is only provided for the conditions on Player 1’s payoffs leading to \(r^*=max\{r',r''\}\). Those conditions ensure that the solution to equation \((\pi _1^3-\pi _1^4)+p(\pi _1^1-\pi _1^2-\pi _1^3+\pi _1^4)=0\), which yields the unique intersection \(\tilde{p}\), is non-positive. Therefore the relevant threshold among \(r'\) and \(r''\) is the largest one.Footnote 18

Take any \(\epsilon >0\), any \([q_{\min },q_{\max }]\subseteq [0,1]\) and any \(\varGamma \in {{\varvec{\Gamma }}}\) such that

  • Player 1’s payoffs are Column Dominant, and

  • the solution to equation \((\pi _1^3-\pi _1^4)+p(\pi _1^1-\pi _1^2-\pi _1^3+\pi _1^4)=0\) is non-positive.

Given Eqs. (1) and (2), we have

$$\begin{aligned} U_1\big (p,[q_{\min },q_{\max }],\epsilon r\big )&=min\big (U_1(p,q_{\min },\epsilon r),U_1(p,q_{\max },\epsilon r)\big )\\&=min\big (U_1(p,q_{\min })+p\epsilon r,U_1(p,q_{\max })+p\epsilon r\big ) \end{aligned}$$

where \(U_1(p,q,\epsilon r)\) is linear in p since \(U_1(p,q)\) is linear in p. For the considered \(\varGamma \), \(\bar{p}\) is not a completely mixed strategy and the unique intersection of \(U_1(p,q_1)\) and \(U_1(p,q_2)\) is in \((\tilde{p},u_1)\) with \(\tilde{p}\le 0\). Therefore, if \(q_{\min }\ne q_{\max }\), we have two possible cases:

  • Case A: \(U_1(p,q_{\min })>U_1(p,q_{\max })\) for all \(p \in (0,1)\),

  • Case B: \(U_1(p,q_{\min })<U_1(p,q_{\max })\) for all \(p \in (0,1)\).

Let \(\hat{q}:=q_{\max }\) if we are in case A and \(\hat{q}:=q_{\min }\) if we are in case B. The mixed strategy \(\hat{q} \in \{q_{\min },q_{\max }\}\) is the one associated with the minimal utility for Player 1, the one she takes into account in front of ambiguity. We have hence

$$\begin{aligned} U_1\big (p,[q_{\min },q_{\max }],\epsilon r\big )&=U_1(p,\hat{q})+p\epsilon r. \end{aligned}$$

From the definition of \(\hat{q}\) and the definition of \(r^*\) in the statement of the lemma, we have \(\epsilon r^*=U_1(0,\hat{q})-U_1(1,\hat{q})\), which can be rewritten \(U_1(0,\hat{q})+0\epsilon r^*=U_1(1,\hat{q})+1\epsilon r^*\) implying that:

  1. 1.

    \(U_1(0,\hat{q})+0\epsilon r>U_1(1,\hat{q})+1\epsilon r\) for all \(r<r^*\),

  2. 2.

    \(U_1(0,\hat{q})+0\epsilon r<U_1(1,\hat{q})+1\epsilon r\) for all \(r>r^*\).

As \(U_1(p,\hat{q})+ p\epsilon r\) is a linear function of p, the best reply to all \(r<r^*\) is \(p=0\) and the best reply to all \(r>r^*\) is \(p=1\). If \(r=r^*\), then \(U_1(p,\hat{q})+p\epsilon r\) is a constant and any \(p \in [0,1]\) is a best reply, and in particular \(p=0\). \(\square \)

For all \(\varGamma \in {{\varvec{\Gamma }}}\) such that Player 2’s payoffs are Row Dominant, there exists similar conditions under which a pure strategy of Player 2 is a best reply to a strategy of Player 1 inducing \([p_{\min },p_{\max }]\).Footnote 19

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Decerf, B., Riedel, F. Purification and disambiguation of Ellsberg equilibria. Econ Theory 69, 595–636 (2020). https://doi.org/10.1007/s00199-019-01186-8

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