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Incomplete Information, Dynamic Stability and the Evolution of Preferences: Two Examples

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Abstract

We illustrate general techniques for assessing dynamic stability in games of incomplete information by re-analyzing two models of preference evolution, the Arce (Econ. Inq. 45(4):708–720, 2007) Employer–Worker game and the Friedman and Singh (Games Econ. Behav. 66:813–829, 2009) Noisy Trust game. The techniques include extensions of replicator and gradient dynamics, and for both models they confirm local stability of the key static equilibria. That is, we obtain convergence in time average for initial conditions sufficiently near equilibrium values.

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Notes

  1. The monomorphic interpretation of a mixture probability q j is that every type j player adopts exactly the same mixed strategy q j W+(1−q j )S. The polymorphic interpretation is that a fraction q j of the type j players adopt the pure strategy W and the rest adopt the pure strategy S. The analysis below works for either interpretation, as well as for the more general interpretation that there is a distribution of pure and mixed strategies among the type j players with overall mean q j .

  2. Here and elsewhere, in modeling the adjustment of shares or mixture probabilities in [0,1] for two (pure) alternatives, there are many smooth monotone (or sign preserving) dynamic specifications to choose among. As catalogued in Weibull [27] and Sandholm [24], these include BNN, perturbed best response, and various sorts of learning dynamics. The techniques illustrated below can straightforwardly be tailored to such specifications. In our experience with state spaces built from [0,1] factors, the stability results are insensitive to the choice of a specific smooth monotone dynamic, but we offer no guarantee.

  3. In the present case, however, we include an extra DE condition in (10) below that \(\dot{\varphi} =0\). This eliminates from the outset those NE for which the two different surviving types of Workers have different material payoffs in equilibrium.

  4. In such cases, it seldom helps to look at second order expansions of the dynamical system, but often third order terms can resolve local stability questions, at the cost of considerable analytic complication. Lyapunov functions are a far more elegant way to establish stability properties, but there is no systematic way of finding such functions. Therefore, as noted below, we favor numerical methods to deal with the problem, and often these methods provide further insights.

  5. Recall from footnote 3 that we only include the NE that have equal payoffs for both Workers’ types when both are present. For instance, the NE (x,p ∗∗,0,(w−m)/(w⋅(1−x))) for x∈[0,m/w] does not satisfy this equal payoff condition.

  6. Stability in time average is also emphasized in the equilibrium concept Time Average of the Shapley Polygon (TASP) proposed by Benaim, Hofbauer and Hopkins [4].

  7. The codes are available upon request.

  8. FS09 argue informally that fitness landscape dynamics will yield degenerate distributions with support on at most two discrete points, one fixed at zero and another at some value v H >c>0 that can vary over time.

  9. We could also have endogenized α in Arce’s model, but that would not have been useful since material payoffs are flat in α except for a discontinuity at a particular threshold that changes type 2 Workers’ behavior. We will see that evolving v makes good sense in the FS09 model.

  10. The evolution of continuous biological traits is commonly modeled via gradient dynamics (e.g., [18, 19, 28]) or by Dieckmann’s restricted version mentioned in the introduction. Continuous strategy sets are seen less often in economics, but there is a cluster of papers beginning with Oechssler and Riedel [20] that applies the continuous extension of the replicator equation. However, economists going back at least to Sonnenschein [25] have also applied gradient dynamics. Friedman and Ostrov [12] argue at length that gradient dynamics are more appropriate when larger changes per unit time are more difficult or expensive, while continuous-state replicator dynamics are more appropriate when adjustment is via deaths and births not spatially connected.

  11. FS09 presents the conditions that x ∗, v ∗, and \(q^{*}_{2}\) should satisfy. In our dynamical system, we can obtain similar conditions considering that the gradient has to be zero for the dynamic equation of v and q 2 and that both types get the same payoff.

  12. \(\hat{e}(k)\) is given by R(k)/(2−2a+2R(k)) where R(k)=(kv(1+v/2)−1)a.

  13. Notice that if both mixing probabilities are pure and q 2−q 1=0, the best reply p is also pure and thus the dynamics is not on a 2D face.

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Acknowledgements

We thank the co-editor Frank Riedel, two anonymous referees, Dann Arce and Bill Sandholm for their comments and suggestions that significantly improved our paper.

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Authors and Affiliations

  1. Department of Economics and Finance, City University of Hong Kong, Tat Chee Ave, Kowloon, Hong Kong

    Jean Paul Rabanal

  2. Economics Department, University of California Santa Cruz, 1156 High Street, Santa Cruz, CA, 95064, USA

    Daniel Friedman

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Appendix: Mathematical Details

Appendix: Mathematical Details

1.1 A.1 Finding DE and NE in the Arce [3] Model

Recall that Sect. 2.4 already identified all corner and edge DE and the subset that are NE.

On the 2D faces that lie inside the 3D faces φ∈{0,1}, Sect. 2.1 noted that the only additional NE are the mixes \((\varphi, p , q_{1} , q_{2} )=(1, \frac{e}{w}, \frac{w-m}{w}, \cdot)\) and \((0,\frac{\alpha-e}{2\alpha-w}, \cdot, \frac{w-m}{w})\). The remaining 2D faces involve φ∈(0,1) and a strict mix of only one of the state variables p,q 1,q 2. The last two cases entail one of the q j pure and the other strictly mixed, but (6) then implies that p is strictly mixed, contradicting the definition of this 2D face. The remaining 2D possibility involves φ,p∈(0,1), which by (7) implies that \(\varphi^{*}=\frac{m/w+q_{2}-1}{q_{2}-q_{1}}\). Ruling out q 2−q 1=0,Footnote 13 we see from (6) that p ∗=e/w. Consequently, the only new candidate equilibria are (φ ∗,p ∗,1,0) and (φ ∗,p ∗,0,1). The dynamics of q 2 depends on the sign of \(\frac{\alpha(w-2e)}{w}\) after plugging p ∗ in (9). The case w−2e>0 is called high incentive wages, and yields the \(q^{*}_{2}=1\) equilibrium, while low incentive wages, the case w−2e<0, yields the equilibrium above with \(q^{*}_{2} = 0\).

We have already found all NE in the 3D faces φ∈{0,1}. The 3D faces p∈{0,1} have no NE, since q j is strictly mixing for states in such faces, and therefore p=p ∗ by (9), contradicting p∈{0,1}. Similarly, the faces q 1∈{0,1} contain no new NE since a strictly mixed strategy for q 2 implies p=p ∗∗=(α−e)/(2α−w) which contradicts the solution of p ∗ in (6). On the faces q 2∈{0,1}, we pick up two new NE, \((\varphi^{*}, \frac{e}{w},\frac{-m+w \varphi^{*} }{w \varphi^{*} } ,1 )\) and \(( \varphi^{*}, \frac{e}{w}, \frac{-m+w}{w \varphi^{*} },0)\); the argument parallels that for the 2D face where φ,p∈(0,1). Keeping the third component q 1∈[0,1] implies the restriction \(\varphi \in [\frac{m}{w}, 1]\) for the first new NE and \(\varphi \in [\frac{w-m}{w}, 1]\) for the second.

Finally, the interior points are unstable since we already know that the dynamics of q 2 depends on the sign of \(\frac{\alpha(w-2e)}{w}\) which forces it to 1 (or zero) in the case of high (or low) wage.

1.2 A.2 Evaluating the Jacobian Matrix at the NE

The text analyzed stability for the first three NE and the last NE listed in (11). In this section, we find Jacobian matrices and eigenvalues for the remaining NE.

The Jacobian matrix for (6)–(9) evaluated at the equilibrium (φ,p,q 1,q 2)=(0,1,⋅,0) is

$$J = \begin{pmatrix} \beta_{\varphi}q_1(w-e) &0&0&0\\ 0&-\beta(w-m) &0&0\\ 0&\beta(1-q_1)q_1 w &\beta (1-2q_1) (w-e) &0\\ 0&0&0&-\beta (\alpha-(w-e)) \end{pmatrix}, $$

whose eigenvalues are {−β(α−(w−e)),−β(w−m),β φ q 1(w−e),β(1−2q 1)(w−e)}. As noted in the text, the first two are always negative in our parameter space. The third is positive except when q 1=0, in which case the last eigenvalue is positive. Hence this NE is definitely not a DSE.

The Jacobian matrix evaluated at (0,0,⋅,1) is

$$J = \begin{pmatrix} \beta_{\varphi}e (1-q_1) &0&0&0\\ 0&-\beta m &0&0\\ 0&\beta(1-q_1) q_1 w &\beta e (-1+2 q_1) &0\\ 0&0&0&-\beta (\alpha-e ) \end{pmatrix}, $$

whose eigenvalues are {−β(α−e),−βm,β φ e(1−q 1),βe(−1+2q 1)}. The third is positive except when q 1=1, in which case the last eigenvalue is positive. Hence this NE also is definitely not a DSE.

The Jacobian at (x,1,1,1) is

$$J = \begin{pmatrix} 0 &0 & \beta_{\varphi} (w-e) (1-\varphi ) \varphi & -\beta_{\varphi}(w-e) (1-\varphi ) \varphi \\ 0 & \beta m & 0 & 0\\ 0 & 0& -\beta(w-e) & 0\\ 0 & 0& 0&\beta (\alpha-(w-e )) \end{pmatrix}, $$

whose eigenvalues are {0,β(α−(w−e)),βm,−β(w−e)}. The second and third are positive, so this equilibrium is not a DSE. Notice that this result also follows from the fact that q 2=1 is not a best-reply for p=1.

The Jacobian at (0,(α−e)/(2α−w),⋅,(w−m)/w) is

$$J = \left( \begin{array}{cccc} -\frac{ \beta_{\varphi} (w-2 e) (m-(1-q_1) w) \alpha }{w (w-2 \alpha )}&0&0&0\\ \frac{-\beta (m-(1-q_1) w) (\alpha-(w-e) ) (\alpha-e ) }{(w-2 \alpha )^2}&0&0&\frac{-\beta w (\alpha-(w-e) ) (\alpha -e) }{(w-2 \alpha )^2}\\ 0&\beta(1-q_1) q_1 w &\frac{ -\beta (-1+2 q_1) (w-2 e) \alpha }{2 \alpha-w }&0\\ 0&-\frac{\beta m (w-m) (2 \alpha-w )}{w^2}&0&0 \end{array}\right), $$

with eigenvalues \(\{\pm\sqrt{\frac{\beta^{2} m (w-m) (\alpha-e ) (\alpha-(w-e)) }{w (2 \alpha-w )}},\frac{\beta (1-2 q_{1}) (w-2 e) \alpha }{2 \alpha-w }, \frac{-\beta(w-2 e) (m-(1-q_{1}) w) \alpha }{w (w-2 \alpha)} \}\). The first pair is real with opposite signs, so this NE is not a DSE. This result is along with the notion that a mixed equilibrium is unstable in the two-population replicator dynamics, see Weibull [27, Chap. 5].

The Jacobian at (1,e/w,(w−m)/w,⋅) is

$$J = \begin{pmatrix} 0&0&0&0\\ \frac{\beta e (w-e) (m+(-1+q_{2}) w) }{w^2}&0&-\beta e (1-\frac{e}{w} ) &0\\ 0&\frac{ \beta m (w-m)}{w}&0&0\\ 0&\beta (1-q_{2}) q_{2} (w-2 \alpha )&0&\frac{\beta (-1+2 q_{2}) (2 e-w) \alpha }{w} \end{pmatrix}, $$

with eigenvalues \(\{0,\pm\frac{\sqrt{-\beta^{2} e m (w-m) (w-e) }}{w},\frac{\beta (-1+2 q_{2}) (2 e-w) \alpha }{w} \}\). The second eigenvalue is imaginary, meanwhile the third can be negative as long as the wage corresponds to the low wage case (w<2e) and q 2<1/2 or the wage is set in the high wage case and q 2>1/2. Hence this NE remains a candidate DSE, requiring further investigation.

The Jacobian at \((\frac{w-m}{w},\frac{e}{w},1,0)\) is

$$J = \begin{pmatrix} 0 &\frac{\beta_{\varphi} m (w-m) }{w} & 0 & 0 \\ -\beta e (1-\frac{e}{w} ) & 0 & \frac{-\beta e (w-e) (w-m) }{w^2} & \frac{-\beta e m (w-e) }{w^2}\\ 0 & 0& 0& 0\\ 0 & 0& 0&\frac{\beta (w-2 e) \alpha }{w} \end{pmatrix}, $$

with eigenvalues \(\{0,\frac{\beta (w-2 e) \alpha }{w},\pm\sqrt{\frac{-\beta_{\varphi} \beta e m (w-m) (w-e) }{w}} \}\). The second is negative in the relevant case of low wages, w−2e<0, and the last pair is pure imaginary. Hence this NE remains a candidate DSE, requiring further investigation. It can be seen to be an extreme case of the NE family listed last in (11) and already analyzed in the text.

At (φ ∗,e/w,0,1), we have φ ∗=m/w and the Jacobian is

$$J = \begin{pmatrix} 0 & -\beta_{\varphi} m (1-\frac{m}{w} ) & 0& 0 \\ \beta e (1-\frac{e}{w} ) & 0 &\frac{-\beta e m (w-e) }{w^2}& \frac{-\beta e (w-e) (w-m) }{w^2}\\ 0 & 0& 0& 0\\ 0 & 0& 0&\frac{-\beta (w-2e) \alpha }{w} \end{pmatrix}, $$

with eigenvalues \(\{0,\frac{-\beta (w-2 e) \alpha}{w},\pm\sqrt{\frac{-\beta_{\varphi} \beta e m (w-m) (w-e) }{w}} \}\). The second is negative in the relevant case of high wages, w−2e>0, so this NE also remains a candidate DSE. It is an extreme case of the next NE family.

The Jacobian at \((\varphi^{*}, \frac{e}{w}, \frac{-m+w \varphi^{*}}{w \varphi^{*}},1 )\) is

$$J = \begin{pmatrix} 0 & -\beta_{\varphi} m (1-\varphi^* ) & 0 & 0 \\ \frac{\beta e m (w-e) }{w^2 \varphi^* } & 0 & -\beta e (1-\frac{e}{w} ) \varphi^* & -\beta e (1-\frac{e}{w} ) (1-\varphi^* ) \\ 0 & \frac{\beta m (-m+w \varphi^* ) }{w \varphi^{*2}} & 0 & 0\\ 0 & 0& 0 & \frac{-\beta (w-2 e) \alpha }{w} \end{pmatrix}, $$

with eigenvalues \(\{0,\frac{-\beta (w-2 e) \alpha }{w},\pm\frac{\sqrt{ \beta e m (w-e) (m (-1+\varphi^{*} )\beta_{\varphi}+(m-w \varphi^{*} ) \beta )}}{w \sqrt{\varphi^{*} }} \}\). The second is negative in the high wage case, and the last pair is pure imaginary since \(\frac{-m+w \varphi^{*}}{w \varphi^{*}}\geq0\), so the entire family with \(\varphi^{*} \in [\frac{m}{w},1]\) is a candidate DSE in the high wage case.

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Rabanal, J.P., Friedman, D. Incomplete Information, Dynamic Stability and the Evolution of Preferences: Two Examples. Dyn Games Appl 4, 448–467 (2014). https://doi.org/10.1007/s13235-013-0096-5

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