6. Equilibrium Payoff Correspondence

In State Space we constructed a partition of the simplex D:= \Delta(\mathcal{Z}). Now, we let D be the domain of the equilibrium payoff correspondence. The task ahead is to approximate the equilibrium value correspondence \mathcal{V}: D \rightrightarrows \mathbb{R}^{\mathcal{Z}} using convex-valued step correspondences.

6.1. Background

Notation reminder:

  • Action profile of small players on [0,1], a \in A. (Assume A is a finite set.) Each small player takes on a personal state j \in \mathcal{Z} := \{ -N,...,-1,+1,...,+M\} at each date t \in \mathbb{N}.

  • Actions of large player (G), b \in B. B := \{b \in \mathbb{R}^{\mathcal{Z}}: -m \leq b(j) \leq \bar{m}, \text{ for } j > 0, \text{ and, } 0 \leq b(j) \leq \bar{m}, \text{ for } j < 0, \forall j \in \mathcal{Z}\} is a finite set and contains vectors b that are physically feasible (but not necessarily government-budget feasible in all states).

  • Extended payoff vector space, \mathbb{R}^{\overline{\mathcal{Z}}}, where \overline{\mathcal{Z}} := \mathcal{Z} \cup \{G\}.

  • Probability distribution of small players on finite set \mathcal{Z}, \lambda \in D := \Delta(\mathcal{Z}).

  • Profile of continuation values of agents, w \in \mathbb{R}^{\mathcal{Z}}.

  • Transition probability matrix at action profile a, P(a)

  • Individual j \in \mathcal{Z}, given action a(j) faces transition probability distribution, p^j(a(j)) \in P(a)

  • Flow payoff profile, v_j(a,b):= u(c^b(j))-\phi(a_j), where

    • v(a,b):=(v_j(a,b))_{j\in \mathcal{Z}};
    • Utility-of-consumption function, u(\cdot); and
    • Disutility-of-effort/action function, -\phi(\cdot).

  • Public date-t history, h^t := \{ \lambda^t, x^t, b^{t-1} \}_{t \geq 0}, where

    • \lambda^t = (\lambda_0,...,\lambda_t) is a history of agent distributions up to and include that of date t;
    • x^t = (x_0,...,x_t) where x_t is a date-t realization of the random variable X_t \sim_{i.i.d.} \mathbf{U}([0,1]); and
    • b^{t-1} = (b_0,...,b_{t-1}) is a history of government policy actions up to the end of date t-1 and let \{b_{-1}\} = \emptyset.

    Also, h^0 := (\lambda_0, x_0).

Definition (Consistency)

Let \mathcal{W}:D \rightrightarrows \mathbb{R}^{\overline{\mathcal{Z}}} be a compact- and convex-valued correspondence having the property that w(G)= \sum_{j\in\mathcal{Z}}\lambda(j)w(j) for all (\lambda,w)\in\text{graph}(\mathcal{W}). A vector (b,a,\lambda',w)\in B \times A \times\Delta(\mathcal{Z}) \times\mathbb{R}^{\overline{\mathcal{Z}}} is consistent with respect to \mathcal{W} at \lambda if

  1. -\lambda b^{T} \geq 0;
  2. \lambda'=\lambda P(a);
  3. w\in\mathcal{W}(\lambda'); and
  4. For all j\in\mathcal{Z}, a(j)\in\text{argmax}_{a'}\left\{(1-\delta) \left[u(c^b(j))-\phi(a')\right]+\delta\mathbb{E}_{p^j(a')}[w(i)]\right \}.

Definition (Admissibility)

For (\lambda,b) \in D \times B let

\pi(\lambda,b):=\min_{(a',\lambda'',w')} \left[(1-\delta)\sum_{j\in\mathcal{Z}}\lambda(j)[u(c^b(j))-\phi(a'(j))] + \delta\sum_{j\in\mathcal{Z}}\lambda''(j)w'(j)\right],

subject to (b,a',\lambda'',w') is consistent with respect to \mathcal{W}(\lambda). Let (\tilde{a}(\lambda,b),\tilde{\lambda}'(\lambda,b),\tilde{w}(\lambda,b)) denote the solutions to the corresponding minimization problem. A vector (b,a,\lambda',w)\in B \times A \times D \times\mathbb{R}^{\overline{\mathcal{Z}}} is said to be admissible with respect to \mathcal{W}(\lambda) if

  1. (b,a,\lambda',w) is consistent with respect to \mathcal{W}(\lambda); and
  2. (1-\delta)\sum_{j\in\mathcal{Z}}\lambda(j)[u(c^b(j))-\phi(a(j))]\delta\sum_{j\in\mathcal{Z}}\lambda'(j)w(j)\geq \max_{b'\in B(\lambda)}\pi(\lambda,b'), where B(\lambda) := \{ b \in B :- \lambda b^{T} \geq 0 \}.

Admissible payoff vectors

The payoff vector defined by an admissible vector (b,a,\lambda',w) at \lambda is given by

E_G(b,a,\lambda',w)(\lambda)&=(1-\delta)\sum_{j\in\mathcal{Z}}\lambda(j)[u(c^b(j))-\phi(a(j))]+ \delta\sum_{j\in\mathcal{Z}}\lambda'(j)w(j) \\ E_j(b,a,\lambda',w)(\lambda)&=(1-\delta) \left[u(c^b(j))-\phi(a(j))\right] + \delta \mathbb{E}_{p^j(a(j))}[w(i)].

Note that E_G(b,a,\lambda',w)(\lambda)=\sum_{j\in\mathcal{Z}}\lambda(j)E_j(b,a,\lambda',w)(\lambda).

In the paper, we proved the following:

SSE Recursive Operator

A SSE is a strategy profile \sigma \equiv \{ \alpha_t, \beta_t\}_{t \geq 0} such that given initial game state \lambda_0, for all dates t \geq 0, and all public histories h^t := \{ \lambda^t, x^t, b^{t-1} \}_{t \geq 0}, a := \alpha_t(h^t, b_t) and b := \beta_t(h^t), and, if \mathcal{V} is the SSE payoff correspondence, then \mathcal{V} is the largest fixed point that satisfies the recursive operator

\mathbf{B}(\mathcal{V})(\lambda):=\text{co}\{E(b,a,\lambda',w)(\lambda)\,\vert\, (b,a,\lambda',w) &\text{ is admissible w.r.t.} \mathcal{V}(\lambda)\},

where \text{co} denotes the convex hull of a set.

The object of interest can be found recursively: \mathcal{V} = \lim_{n \rightarrow +\infty} \mathbf{B}^n(\mathcal{W}_0), for any initial convex-valued and compact correspondence \mathcal{W}_0.

Note

Given state \lambda and agent payoff vector w \in \mathbb{R}^\mathcal{Z} determine a unique corresponding government payoff given by \lambda \cdot w. We can thus ignore the government payoff when defining the equilibrium value correspondences and their approximations, and restrict their codomain to \mathbb{R}^\mathcal{Z}.