# Positional 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 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-03947632

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

Distributed under a Creative Commons Attribution 4.0 International License
Positional 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.
Standard assumption

"Each individual's preference is independent of the behavior of other agents"$\nu$

Preferences are interdependent.

Classical literature:

* Duesenberry (1949). Relative income hypothesis

* Leibenstein (1950). Comparing consumption

* **Conformism.** Bandwagon effect. Agents follow the consumption behavior of others.

* **Snobism.** Agent's desire for exclusiveness.
Influence of social context central in modern behavioral econ.

<figure><img src="image_2.png" /></figure>

**Red Queen effect:** Negative consumption externality

Race in conspicuous consumption. Too much expending

Inefficiency → Loss in Social welfare
Research question

What if the positional good is contribution to a public good?

i) Positional concerns $\Rightarrow$ positive contributions?
if so ...

ii) would it necessary lead to social welfare improvements?

Related literature:

* Bougherara et al. (2019). Same research question.
We propose more systematic generalization.
* The static model. Two-player public good game:

* The dynamic model with inertia.

* Conclusions & Extensions
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 contribution)

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

* 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.

$$ C2a: \frac{\partial u}{\partial x_i}(0, 0) > 0, \quad C2b: \frac{\partial u}{\partial x_i}(w_1, w_1 + w_2) < 0. $$

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

C3 Agents contributions are substitutes

$$ C3: \frac{\partial^2 u}{\partial x_i \partial X}(x_i, X) < 0. $$
* The intrinsic utility is an additively separable function:

$$u_i(x_i, X) = w_i - x_i + b_i(X), \quad u(X) = w_1 + w_2 - X + b_1(X) + b_2(X).$$

<figure><img src="image_3.png" /></figure>

$\alpha_i$ PG valuation by $i$,
$\varepsilon$ satiation

* Positional payoff

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

Positional concern of player $i$: $v_i^P \ge 0$
The problem for player $i \in \{1, 2\}$ is:

$$ \max_{0 \le x_i \le w_i} w_i - x_i + b_i(X) + v_i^P (x_i - x_j). $$

The marginal utility of player $i$ now reads as:

$$ \frac{\partial U_i}{\partial x_i} = -1 + b'_i(X) + v_i^P. $$

Positionality raises the marginal benefit of private provision. Private provision can become individually rational for player $i$.
The FOC condition: $b'_i(X) = 1 - v_i^P$.

In equilibrium,

<figure><img src="image_4.png" /></figure>

$v_i^P < 1 \rightarrow b'_i(X) > 0$: **PG.**

Assumption $0 \le v_i^P < 1$

<figure><img src="image_5.png" /></figure>

$v_i^P > 1 \rightarrow b'_i(X) < 0$: **PB.**

## 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 $v_i^P > 1 - \alpha_i$
a) If $w_1 + w_2 \le \min\{A_1, A_2\}$, then $(x_1^N, x_2^N) = (w_1, w_2)$.

b) If $w_1 + w_2 > \min\{A_1, A_2\}$ and

bI) $A_1 = A_2 = A > 0$, set of Nash equilibria:

$$ \mathcal{N} = \{ (x_1, x_2) | x_1 + x_2 = A \wedge (x_1, x_2) \in [0, \min\{A, w_1\}] \times [0, \min\{A, w_2\}] \} $$

bII) If $A_i > A_j$ and $A_i > 0$, $i, j \in \{1, 2\}$, $i \ne j$,

$$ (x_i^N, x_j^N) = \begin{cases} (w_i, A_j - w_i) & w_i < A_j < w_i + w_j \quad [\text{bII.1}], \\ (w_i, 0) & A_j \le w_i < A_i \quad [\text{bII.2}], \\ (A_i, 0) & A_i \le w_i \quad [\text{bII.3}]. \end{cases} $$

c) $A_1 = A_2 = 0$: $(x_1^N, x_2^N) = (0, 0)$.
bII.1: $(w_i, A_j - w_i)$, bII.2: $(w_i, 0)$, bII.3: $(A_i, 0)$

Case of symmetric valuation of PG consumption, $\alpha_i = \alpha_j = \alpha$.

<figure><img src="image_6.png" /><figcaption>Figure: Level curves for $u(X): \varepsilon = 0.7$ (left), $\varepsilon = 0.4$ (right).</figcaption></figure>
bII.1: $(w_i, A_j - w_i)$, bII.2: $(w_i, 0)$, bII.3: $(A_i, 0)$

Case of symmetric valuation of PG consumption, $\alpha_i = \alpha_j = \alpha$.

<figure><img src="image_7.png" /><figcaption>Figure: Level curves for utility: $\epsilon = 0.7$ (left), $\epsilon = 0.4$ (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 $t$ a **positional** agent gets joy from contributions above the other player at time $t-1$: $+v_i^P(x_{it} - x_{jt-1})$.

2. Agents have **Inertia** from previous actions. Disutility from deviations from the previous action: $-v_i^I(x_{it} - x_{it-1})^2/2$.

Utility at time $t$:

$$ 
\begin{aligned}
U_i(\bullet) = & w_i - x_{it} + \alpha_i \left[ x_{it} + x_{jt} - \frac{\varepsilon}{2} (x_{it} + x_{jt})^2 \right] \\
& + v_i^P (x_{it} - x_{jt-1}) - \frac{v_i^I}{2} (x_{it} - x_{it-1})^2.
\end{aligned}
$$
We consider the intrinsic utility function of the static model...

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

2. Agents have **Inertia** from previous actions. Disutility from deviations from the previous action: $-v_i^I(x_{it} - x_{it-1})^2/2$.

Utility at time $t$:

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

Alternative dynamics based completely in the present (except for inertia) or completely in the past → no big differences.
The problem of a myopic agent $i \in \{1, 2\}$ is:

$$ \begin{array}{l} \max_{x_{it}} U_i(x_{it}, x_{jt}, \hat{x}_{it}, \hat{x}_{jt}), \\ \text{s.t.:} \quad \hat{x}_{it} = x_{it-1}, \quad i \in \{1, 2\}. \end{array} $$

The first-order conditions for an interior equilibrium are:

$$ -1 + \alpha_i [1 - \varepsilon(x_{it} + x_{jt})] - v_i^l(x_{it} - \hat{x}_{it}) + v_i^p, = 0 \quad \forall i \in \{1, 2\}. $$
$$ \begin{pmatrix} x_{it} \\ x_{jt} \end{pmatrix} = \delta t \begin{pmatrix} 1 \\ -1 \end{pmatrix} + \begin{pmatrix} x_{i0} \\ x_{j0} \end{pmatrix} + (1 - \lambda_2^t) \frac{X^* - X_0}{\alpha_i v_j^! + \alpha_j v_i^!} \begin{pmatrix} \alpha_j v_i^! \\ \alpha_i v_j^! \end{pmatrix}, $$

where,

$$ \delta = \frac{\varepsilon \alpha_i \alpha_j (A_i - A_j)}{\alpha_i v_j^! + \alpha_j v_i^!} \quad \lambda_2 = \frac{v_i^! v_j^!}{v_i^! v_j^! + \varepsilon (\alpha_i v_j^! + \alpha_j v_i^!)} $$

$$ X^* = \frac{\alpha_i v_j^! A_i + \alpha_j v_i^! A_j}{\alpha_i v_j^! + \alpha_j v_i^!}. $$

$$ X_t = X_0 + (1 - \lambda_2^t)(X^* - X_0). $$
Myopic dynamic problem: $X_0 = 0$ and $A_i \ge A_j$

Asymptotic behavior of the sequence of contributions:

* Static equilibrium: a fixed point of dynamical system,

* trajectory of contributions hits the boundary of feasible contributions, where the static equilibrium lies ...

* ...then converges monotonically to this static equilibrium.

* Without inertia, convergence in one step.
Myopic dynamic problem: $X_0 = 0$ and $A_i \ge A_j$

Asymptotic behavior of the sequence of contributions:

* Static equilibrium: a fixed point of dynamical system,

* trajectory of contributions hits the boundary of feasible contributions, where the static equilibrium lies ...

* ...then converges monotonically to this static equilibrium.

* Without inertia, convergence in one step.

Possible overshooting for player $j$: contribution increases initially and decreases later on.

* Overshooting for player $j$ can lead to overshooting of total contribution when the player who wishes the most contributes her total endowment (cases [bII.1] and [bII.2])
Simulations: bll.1 ($w_i$, $A_j - w_i$) and bll.2 ($w_i$, 0)

$$ \text{bll.1} \quad x^N = (0.35, 0.04), \quad x^{SO} = 0.416, \quad x^{E0} = .83 $$

<figure><img src="image_8.png" /><figcaption>Figure: Individual (left) and total (right) contributions</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure: Intrinsic utility (left) and social welfare (right)</figcaption></figure>
Simulations: bll.1 ($w_i$, $A_j - w_i$) and bll.2 ($w_i$, 0)

<figure><img src="image_10.png" /><figcaption>Figure: Intrinsic utility (left) and social welfare (right)</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure: Trajectories for two symmetric players</figcaption></figure>
$$ \text{bII.3} \quad x^N = (0.75, 0), \quad x^{SO} = 0.62, \quad x^{E0} = 1.25 $$

<figure><img src="image_12.png" /><figcaption>Figure: Individual and total contributions</figcaption></figure>
1. Introduction

2. Static game

3. Dynamic Game

4. Conclusions and extensions
In the static framework

* Nash equilibrium for an additive separable public good game with positional concerns.

* Different equilibria depending on players positional concern (wished amounts) vs endowments.

* Positional concerns, contributions and social welfare.

In the myopic dynamic framework with inertia

* Convergence to the static case

* Possibility of overshooting in individual and total contribution.
* A positional agent versus a conformist (joy from doing like others). Disutility: $-v_i^c(x_{it} - x_{jt-1})^2/2$.

* Consider another dynamic: the stock of public good.

* Consider farsighted agents (true dynamic game).

* More than two players.

* ...
Obrigado!
Proposition

Assume $w_i = w$ and $\alpha_i = \alpha$. $\hat{\varepsilon} = \frac{2\alpha-1}{w\alpha}$. Then:

1. If $\varepsilon < \hat{\varepsilon}$ (low satiation), IW-I area wider than IW-R area.

2. If $\varepsilon > \hat{\varepsilon}$ (high satiation), IW-R area wider than IW-I area.
Bougherara et al. (2019): What if positional concerns of the two player are both higher?

The effect of a joint increment in the positional concerns of both players is (in bll1 and bll3):

$$ \frac{dU^N}{dv_i^P} + \frac{dU^N}{dv_j^P} \ge 0 \Leftrightarrow v_i^P + v_j^P \le 1. $$
Simulations: bll.1 ($w_i$, $A_j - w_i$) and bll.2 ($w_i$, 0)

$$ \text{bll.1} \quad x^N = (1, 0.22), \quad x^{SO} = 0.62, \quad x^{E0} = 1.25 $$

<figure><img src="image_13.png" /><figcaption>Figure: Individual (left) and total (right) contributions</figcaption></figure>
Simulations: bll.1 ($w_i$, $A_j - w_i$) and bll.2 ($w_i$, 0)

<figure><img src="image_14.png" /><figcaption>Figure: Intrinsic utility (left) and social welfare (right)</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure: Location of the Nash equilibrium, when $A_i > A_j > 0$</figcaption></figure>