# Positional and conformist effects in public good provision. Strategic interaction and inertia

Francisco Cabo, Alain Jean-Marie, Mabel Tidball

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Francisco Cabo, Alain Jean-Marie, Mabel Tidball. Positional and conformist effects in public good provision. Strategic interaction and inertia. 19th International Symposium on Dynamic Games and Applications, Mark Broom, Alberto A. Pinto, Jul 2022, Porto, Portugal. hal-03947724

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Positional and conformist effects in public
good provision. Strategic interaction and
inertia

Francisco Cabo, IMUVa, Universidad de Valladolid, Spain

Alain Jean-Marie, INRIA, Sophia Antipolis, France

Mabel Tidball, INRA, CEEM, Montpellier, France

19th International Symposium on Dynamic Games and Applications.
July 25-28 2022.
In the previous presentation

*Positional effects in public good provision.*

*Strategic interaction and inertia*

we considered a game of contribution to a public good, in which players are Positional: concerned with their relative contribution.

i) Positional concerns ⇒ positive contributions?
Yes!

ii) Does it necessary lead to social welfare improvements?
No!

iii) If it does, can inertia in adjustments harm welfare?
Yes!

Concluded with possible extension: what if one player is Positional and one is Conformist?
A public-good game with subjective effects: one player Positional, one player Conformist.

* The static model.

* Nash Equilibrium

* The dynamic model with inertia.

* Some theoretical results.

* Numerical simulations.

* Conclusions
1. Introduction

2. Static game

3. Dynamic Game

4. Conclusions and extensions
* Two players.

* Each player endowed with $w_i$ and contributes $x_i \in [0, w_i]$ to a public good. Then $w_i - x_i$ privately consumed.

* Preferences:
  **intrinsic utility** (absolute level of contribution)
  **positional payoffs** (associated with relative contrib.)

$$U_i(x_i, x_j) = u_i(x_i, X) + V_i(x_i - x_j), \quad X = x_i + x_j$$

**conformist payoffs** (associated with distance to contrib.)

$$U_j(x_i, x_j) = u_j(x_j, X) + V_j(x_i - x_j).$$

* Global intrinsic utility: $u = u_i + u_j$

* The global utility or social welfare: $U = U_i + U_j$
C1 Individual provision always reduces own welfare:

$$ \frac{\partial u_i}{\partial x_i}(x_i, X) < 0, \quad \forall (x_i, x_j) \in [0, w_i] \times [0, w_j]. $$

Nash equilibrium is (0, 0).

C2 Individual provision: First unit ↑ $u$; last unit ↓ $u$.

$$ C2a: \frac{\partial u}{\partial x_i}(0, 0) > 0, $$

$$ C2b: \frac{\partial u}{\partial x_i}(w_1, w_1 + w_2) < 0. $$

Some contribution to the public good and to the private good are socially desirable.

C3 Agents contributions are substitutes

$$ C3: \frac{\partial^2 u}{\partial x_i \partial X}(x_i, X) < 0. $$
Functional specifications:

* The intrinsic utility is an additively separable function:

$$u_i(x_i, X) = w_i - x_i + b_i(X)$$

$$u(X) = w_1 + w_2 - X + b_1(X) + b_2(X)$$

$$b_i(X) = \alpha_i \left( X - \frac{\varepsilon}{2} X^2 \right)$$

Public Good valuation by i: $\alpha_i$, satiation $\varepsilon$

* Positional payoff

$$v_i(x_i - x_j) = v_i^P \times (x_i - x_j).$$

Positional concern of player i: $v_i^P \ge 0$

* Conformist payoff

$$v_j(x_i - x_j) = - \frac{v_j^C}{2} \times (x_i - x_j)^2.$$

Conformist concern of player j: $v_j^C \ge 0$
The problem for player $k \in \{i, j\}$ is:

$$ \begin{aligned} \max_{0 \le x_i \le w_i} & \quad w_i - x_i + b_i(X) + v_i^P (x_i - x_j) \\ \max_{0 \le x_j \le w_j} & \quad w_j - x_j + b_j(X) - \frac{v_j^C}{2} (x_i - x_j)^2 \end{aligned} $$

The marginal utility of players $i$ and $j$ now reads as:

$$ \begin{aligned} \frac{\partial U_i}{\partial x_i} &= -1 + b'_i(X) + v_i^P \\ \frac{\partial U_j}{\partial x_j} &= -1 + b'_j(X) + v_j^C(x_i - x_j) \end{aligned} $$

Positionality raises the marginal benefit of private provision. Conformism does also, as long as $j$ contributes less than $i$.
For the positional player, a similar situation:

Best-reply function:

$$x_i^b(x_j) = \begin{cases} 0 & A_i \le x_j \\ A_i - x_j & A_i - w_i \le x_j \le A_i \\ w_i & x_j \le A_i - w_i. \end{cases}$$

Definition (Wished amount)

$$A_i = \begin{cases} 0 & \text{if } v_i^P < 1 - \alpha_i, \\ \frac{v_i^P - (1 - \alpha_i)}{\alpha_i \varepsilon} & \text{otherwise} \\ w_i + w_j & \text{if } v_i^P > 1 - \alpha_i + \alpha_i \varepsilon (w_1 + w_2). \end{cases}$$

Positive contribution $v_i^P > 1 - \alpha_i$

Public Good Assumption $v_i^P < 1$
For the conformist player, different situations

Referring to condition C3:

$$ \frac{\partial^2 U_j}{\partial x_i \partial x_j} = -\alpha_j \varepsilon + v_j^c $$

1. $v_j^c < \alpha_j \varepsilon$: contributions are still substitutes

2. $v_j^c > \alpha_j \varepsilon$: conformism is so strong that contributions from the other agent increases the willingness to contribute: contributions become complements for Agent j.

Best-reply function (complementarity case):

$$ x_j^b(x_i) = \begin{cases} 0 & x_i \le B_j := \frac{1 - \alpha_j}{v_j^c - \alpha_j \varepsilon} \\ \frac{v_j^c - \alpha_j \varepsilon}{v_j^c + \alpha_j \varepsilon} (x_i - B_j) & \text{otherwise} \\ w_j & w_j(v_j^c + \alpha_j \varepsilon) \le (x_i - B_j)(v_j^c - \alpha_j \varepsilon) \end{cases} $$
Unique Nash equilibrium in all cases.

c) If $v_i^P \le 1 - \alpha_i$, $A_i = 0$: no contribution at all

$$ (x_i^N, x_j^N) = (0, 0) $$

a) If $v_i^P > 1 - \alpha_i$, $A_i > 0$, and

$$ v_j^C \le \underline{v_j^C} := \alpha_j \varepsilon + \frac{1 - \alpha_j}{\min\{A_i, w_i\}} $$

then

$$ (x_i^N, x_j^N) = (\min\{A_i, w_i\}, 0) $$

Agent *j* is not incentivized enough to contribute. This includes the substituability case $v_j^C \le \alpha_j \varepsilon$, but also the **weak complementarity** case $v_j^C \in (\alpha_j \varepsilon, \underline{v_j^C}]$.
b) If $v_i^P > 1 - \alpha_i$, $A_i > 0$, and $v_j^C > \underline{v}_j^C$ (strong complementarity)

b$_{int}$) unique interior equilibrium

$$x_{i \ int}^{\text{N,PC}} = \frac{1}{2} (A_i + \Delta_x) \qquad x_{j \ int}^{\text{N,PC}} = \frac{1}{2} (A_i - \Delta_x)$$

$$\Delta_x := \frac{\alpha_i - \alpha_j(1 - v_i^P)}{\alpha_i v_j^C}$$

b$_1$) $(w_i, x_j^b(w_i))$

b$_{12}$) $(w_i, w_j)$

b$_2$) $(A_i - w_j, w_j)$
Nash Equilibrium (end)

<figure><img src="image_2.png" /><figcaption>Nash equilibria with positive contribution
for $w_1 > w_2$ (left); $w_1 \le w_2$ (right)</figcaption></figure>
1. Introduction

2. Static game

3. Dynamic Game

4. Conclusions and extensions
We consider the intrinsic utility function of the static model...

1. At time $t$ the **positional** agent gets joy from contributions above the other player at time $t-1$: 
   $$+v_i^P(x_{it} - x_{jt-1}).$$

2. The **conformist** agent gets joy from contributions close to the other player at time $t-1$: 
   $$-v_j^C(x_{jt} - x_{it-1})^2/2.$$

3. Agents have **Inertia** from previous actions. Disutility from deviations from the previous action: 
   $$-v_k^I(x_{kt} - x_{kt-1})^2/2, k \in \{i,j\}.$$
Utilities at time t:

$$U_i(\bullet) = w_i - x_{it} + \alpha_i \left[ x_{it} + x_{jt} - \frac{\varepsilon}{2}(x_{it} + x_{jt})^2 \right] + v_i^P (x_{it} - x_{jt-1}) - \frac{v_i^I}{2}(x_{it} - x_{it-1})^2$$

$$U_j(\bullet) = w_j - x_{jt} + \alpha_j \left[ x_{it} + x_{jt} - \frac{\varepsilon}{2}(x_{it} + x_{jt})^2 \right] - \frac{v_j^C}{2}(x_{jt} - x_{it-1})^2 - \frac{v_j^I}{2}(x_{jt} - x_{jt-1})^2$$
The problem of a myopic agent $k \in \{i, j\}$ is:

$$ \begin{array}{l} \max_{x_{kt}} U_k(x_{kt}, x_{-kt}, \hat{x}_{kt}, \hat{x}_{-kt}), \\ \text{s.t.:} \quad \hat{x}_{kt} = x_{kt-1}. \end{array} $$

The first-order conditions for an interior equilibrium are:

$$ \begin{pmatrix} \varepsilon\alpha_i + v_i^l & \varepsilon\alpha_i \\ \varepsilon\alpha_j & \varepsilon\alpha_j + v_j^l + v_j^c \end{pmatrix} \begin{pmatrix} x_{it} \\ x_{jt} \end{pmatrix} = \begin{pmatrix} v_i^l & 0 \\ v_j^c & v_j^l \end{pmatrix} \begin{pmatrix} x_{it-1} \\ x_{jt-1} \end{pmatrix} + \begin{pmatrix} -1 + \alpha_i + v_i^p \\ -1 + \alpha_j \end{pmatrix}. $$

⇒ dynamic system $x_t = Mx_{t-1} + V_0$.
Properties of the free dynamical system

● When $v_k' \neq 0$ at least for one player $k \in \{i, j\}$,

► both eigenvalues have modulus < 1

► when they are real, they have the same sign

► the dynamical system $x_t$ converges to a unique steady state which is the point $x_{int}$

● When $v_i' = v_j' = 0$

► one eigenvalue is equal to -1, the other one is 0

► the dynamical system does not converge in general: its trajectories tend asymptotically to a cycle of order 2 which is organized around the point $x_{int}$
# General properties (continued)

Behavior different from the Positional+Positional case

## Eigenvalues

A sufficient condition for both eigenvalues to be complex is:

$$v_i^l < \frac{\alpha_i}{\alpha_j} (v_j^l + v_j^c)$$

Spiralling behavior:

* small inertia of Positional player

* large inertia or conformism from Conformist player

When taking budget constraints into account:

## Property of the constrained dynamical system

If $x_0 = (0, 0)$, then $x_1 = (x_{11}, 0)$.
Illustration 1: no inertia

Parameters: $\epsilon = 0.4$, $\alpha_1 = 0.8$, $\alpha_2 = 0.5$, $v_1^P = 0.56$, $v_2^C = 2$.
Wished amount $A_1 = 1.125$, $A_1/2 = 0.5625$.

<figure><img src="image_3.png" /></figure>
Illustration 2: spiral or direct convergence

Varying inertia parameters:

* case 1: $v_1' = 10, v_2' = 10: \lambda_1 \approx 0.8543, \lambda_2 \approx 0.9302$

* case 2: $v_1' = 5, v_2' = 10: \lambda_i \approx 0.8757 \pm 0.0655i$

* case 3: $v_1' = 20, v_2' = 20: \lambda_i \approx 0.8201 \pm 0.1068i$

<figure><img src="image_4.png" /></figure>
Welfare analysis: remember the typology of total contribution X:

<figure><img src="image_5.png" /></figure>
Illustration 2: welfare analysis (continued)

<figure><img src="image_6.png" /></figure>
Varying inertia parameters:

* case 1: $v_1' = 0.1, v_2' = 0.5$: $\lambda_1 \approx -0.7392, \lambda_2 \approx -0.0025$

* case 2: $v_1' = 0.1, v_2' = 5$: $\lambda_1 \approx -0.4705, \lambda_2 \approx -0.0337$

* case 3: $v_1' = 1, v_2' = 1$: $\lambda_i \approx 0.0848 \pm 0.1024i$

<figure><img src="image_7.png" /></figure>
1. Introduction

2. Static game

3. Dynamic Game

4. Conclusions and extensions
With respect to the Positional+Positional case, there are common features and qualitative differences.

In the static framework

≠ The standard situation with large endowments is an *interior Nash equilibrium*

In the myopic dynamic framework with inertia

= Convergence to the static equilibrium

≠ Possibility of spiralling or oscillating convergence

= Possibility of overshooting

= Possibility of transient welfare reduction
As in the previous talk:

* Consider another dynamic: the stock of public good.

* Consider farsighted agents (true dynamic game).

* More than two players.

* ...
Obrigado!