Political Manifestos

GameTheory Exercise

Manifesto: 宣言

This is a problem set question from EC487 Advanced Microeconomics, related to the Game Theory stuff.

Question

Three players, two politicians $j\in \{1,2\}$ and one voter, play an extensive-form game:

• stage 1 each politician $j\in \{1,2\}$ simultaneously proposes an electoral manifesto $M^j\in \{\{x\},\{x,y\}\}\equiv \mathcal{M}$
• stage 2 the voter elects one of the two politicians $j\in \{1,2\}$
• stage 3 nature draws a payoff-relevant state $\theta \in \{x,y\}\equiv \Theta$ that is equal to $y$ with probability $p:0<p<\frac{1}{2}$ and $x$ with probability $1-p$
• stage 4 the elected politician $j$ chooses a policy $a\in M^j\subset A\equiv \{x,y\}$

Preferences over terminal histories $(M^1,M^2,j,\theta ,a)$ are as follows:

• voter utility is one if $a=\theta$, zero otherwise
• elected politician utility is $b+1$ if $a=y$ and equal to $b>0$ otherwise
• non-elected politician utility is one if $a=y$ and zero otherwise

Having studied the stages 1-4 game in the lecture, we now consider the complete-information infinite horizon game that repeats stages 1-4 each period. Stage game utility is discounted at rate $\delta$ and normalized by $1-\delta$. E.g., politician utility when elected and choosing policy $y$ each period is

$(1-\delta ){\sum }_{k=0}^{\infty }{\delta }^k(b+1)=(b+1).$

The Tasks

1. Characterize the set of $(\delta ,b,p)$ for which there exists an SPE of the repeated game in which on-path elected politicians play $a=x$.

We first clarify what is on-path and what is off-path here. On-path means that the actions that are taken according to the equilibrium strategies, while off-path means actions that deviate from the equilibrium strategies.

Before answering the question, I would want to discuss more about the stage 3: what is the nature draws a payoff-relevant state $\theta$. Here, $\theta$ means a specific state that what the voter prefers. The voter prefers policy $x$ when $\theta =x$, and prefers policy $y$ when $\theta =y$. It’s not a proportion, it’s just a state that determines the voter’s preference.

Story

In this question, since the on-path elected politicians play $a=x$, every politician $j$ would proposes $M^j=\{x\}$. Without generality, lets say at stage 2, the voter chooses politician 1. At stage 3, it doesn’t matter what $\theta$ is since the elected politician would always choose $a=x$ at stage 4.

Using the backward induction to verify it.

• At stage 4, since $M^j=\{x\}$, you only play $a=x$. The elected politician 1 gets utility $b$, the non-elected politician 2 gets utility $0$, and the voter gets utility $1$.
• At stage 3, No action involved.
• At stage 2, the voter would choose either politician since both would propose $M^j=\{x\}$. The voter gets utility $1$ regardless of the choice.

We think about the motivation to deviate. When does the penalty happens? It happens at stage 1 when a politician $j$ proposes a manifesto with $y$ which is $\{x,y\}$ here. The action of the voter is not to vote him and vote the other one. Thus it’s natural that such $X_j$ is a SPE.

2. Characterize the set of $(\delta ,b,p)$ for which there exists an SPE of the repeated game in which on-path elected politicians play $a=y$.

3. Characterize the set of $(\delta ,b,p)$ for which there exists an SPE of the repeated game in which on-path elected politicians match the state $a=\theta$.

4. Which policy is easier to sustain on-path in an SPE of the repeated game: matching the (voter-preferred) state $a=\theta$ or selecting the (politician-preferred) policy $a=y$?