Problem 224 - An Introduction to Game Theory

Question

The following pair of games is designed to study the implications of different legislative procedures for the cohesion of a governing coalition. In both games a legislature consists of three members. Initially a governing coalition, consisting of two of the legislators, is given. There are two periods. At the start of each period a member of the governing coalition is randomly chosen (i.e. each legislator is chosen with probability \(\frac{1}{2}\) ) to propose a bill, which is a partition of one unit of payoff between the three legislators. Then the legislators simultaneously cast votes; each legislator votes either for or against the bill. If two or more legislators vote for the bill, it is accepted. Otherwise the course of events differs between the two games. In a game that models the current US legislature, rejection of a bill in period \(t\) leads to a given partition \(d^{t}\) of the pie, where \(0

Step-by-Step Solution

Answer

US: Optimal strategy considers future payoffs; UK: Optimal strategy avoids government loss while maximizing payoff.

Step 1 - Understand the Problem

The problem involves two legislative procedures (US and UK) and requires finding the subgame perfect equilibrium outcomes for both. These procedures involve how bills are proposed, voted on, and the consequences of their rejection.

Step 2 - Define the Game for the US Legislature

In the US model, if the bill is rejected in period 1, the division of the payoffs follows a given partition \( d^t \).

Step 3 - Define the Game for the UK Legislature

In the UK model, rejection of a bill in period 1 results in the dissolution of the government, selection of a new governing coalition, and no legislator receives any payoff in that period.

Step 4 - Analyzing Subgame Perfect Equilibrium for US

For the US model, if a bill is rejected in period 1, the same governing coalition proposes another bill in period 2. Since each proposer aims to maximize their own payoff within the constraints, we must calculate the equilibrium considering payoffs in subsequent periods, denoted as \(d^t\).

Step 5 - Analyzing Subgame Perfect Equilibrium for UK

In the UK model, the key factor is that rejection results in a loss of payoff for the period and the formation of a new coalition. Consequently, each member of the coalition considers the potential loss when deciding whether to accept or reject a bill.

Step 6 - Calculate Payoffs and Strategies for US

For each legislator in the US model, consider the probabilities of being chosen as proposer, accepting payoffs, and the iterations over periods. Form the optimal strategy set to maximize each legislator's total expected payoff over two periods.

Step 7 - Calculate Payoffs and Strategies for UK

In the UK model, each coalition member compares the immediate payoff from accepting the current bill versus the expected value of future proposals with a new coalition. Formulate the strategies to avoid government dissolution while maximizing payoffs.

Step 8 - Determine Subgame Perfect Equilibrium Outcomes

For both US and UK models, combine the optimal strategies of all players. The subgame perfect equilibrium is where no player has an incentive to deviate from their strategy given the equilibrium strategies of others.

Key Concepts

Legislative Procedures

To understand legislative procedures in the context of game theory, we recognize it as a set of rules governing the proposal, voting, and consequences of legislative actions. In our exercise, these rules differ for the US and UK legislatures. In the US case, rejection of a bill leads to a defined partition of payoffs, and the existing governing coalition remains intact. Conversely, in the UK system, rejecting a bill causes the government to collapse, leading to a new coalition and no payoffs in that period. This distinction significantly impacts the strategies of the legislators during the game. The procedure determines how proposals are made and how outcomes are enforced, which directly influences the behavior of lawmakers.

Game Theory

Game theory is the study of strategic interactions where the outcome for each participant depends on the choices of all involved. Here, we are dealing with a legislative game involving three lawmakers. They must decide on proposals and vote, considering possible reactions and consequences. The subgame perfect equilibrium (SPE) is a refinement of Nash equilibrium used here. It's where players make optimal decisions at every stage, including off-the-equilibrium paths. By dissecting the game into smaller subgames and ensuring optimal strategies at each stage, we identify the SPE, ensuring robust and realistic outcomes in complex legislative settings.

Proposer Selection

Proposer selection is crucial in our legislative models. In both the US and UK models, the proposer is chosen randomly from the governing coalition. With each legislator having a 50% chance of being selected, this randomization introduces an element of unpredictability. Each member of the coalition must prepare for any scenario, as becoming the proposer gives significant control over the payoff distribution. Effective strategies thus involve not just maximizing current payoffs but also anticipating future opportunities to propose a bill, making proposer selection a pivotal element of their strategic planning.

Payoff Distribution

Payoff distribution in our exercise refers to how the one unit of payoff is divided among the legislators. The proposed partition is crucial since it directly impacts the acceptance of the bill. In the US model, a predefined partition occurs if a bill is rejected, influencing legislators’ decisions. In contrast, the UK model results in no payoff if the bill is rejected, thus putting pressure on legislators to compromise and accept the bill to avoid getting nothing. Understanding how different distributions affect acceptance helps in formulating strategies aligning with legislators' goals of maximizing their share.

Coalition Stability

Coalition stability is the ability of the governing coalition to remain intact throughout the legislative process. This stability is tested when bills are proposed and voted upon. In the US system, coalition stability is high because the same coalition proposes even after a bill’s rejection. However, in the UK model, the rejection of a bill leads to the collapse of the government and formation of a new coalition. This dynamic forces legislators to weigh the stability of their coalition against immediate payoffs. Stable coalitions are more likely to pass bills, while unstable ones face rejection and constant shifts, affecting overall legislative efficiency.