# Game Theory and Incentives#

## Game Theory and Mechanism Design#

Classical game theory analyzes strategic behavior:

Given a strategic environment (a “game”), what can we say about the likely outcome?

Mechanism design flips the problem around

How can we create rules of play (a “mechanism”) so that even if players are strategically sophisticated, we still get a desired outcome?

- game#
A game in a strategic form is a triple \((N, A, u)\) consisting of:

A set of players \(N = \{1, \ldots, n\}\)

A set of possible

*actions*\(A_i\) for player \(i\) for \(i \in N\)Payoff functions for each player: \(u_i(a_1, \ldots, a_n)\) for \(u: A_1 \times \dots \times A_n \to \mathbb{R}\)

- weakly dominant action#
Action \(a_i\) is weakly

*dominant*for player \(i\) if for all \(a_i'\) and \(a_{-i}\): \(u_i(a_i, a_{-i}) \geq u_i(a_i', a_{-i})\) and for some \(\hat{a}_{-i}\), \(u_i(a_i, \hat{a}_{-i}) > u_i(a_i', \hat{a}_{-i})\).

Tip

Intuition
It’s a no-brainer for \(i\) to play a dominant action if she has one; it is an optimal choice *regardless of how the others may play*.

- Nash equilibrium#
Action profile \(a = (a_1, \ldots, a_n)\) is a

*Nash equilibrium*if for each player \(i\) and action \(a_i'\), \(u_i(a_i, a_{-i}) \geq u_i(a_i', a_{-i})\). In words, an action profile \(a\) is a Nash equilibrium if no player \(i\) can increase its own payoff above \(u_i(a)\) by unilaterally changing in its action from \(a_i\) to some other action \(a_i'\).

### Incomplete Information#

What happens if the players don’t know each other’s payoff functions? For example, in an auction, players may not know the other bidders’ valuations.

(Bayesian) Nash equilibrium is more subtle: each player’s optimal choice depends on what it believes others’ payoffs may be, so each player’s expected payoff depends on probabilities of what others would do with different payoffs.

### Allocation Problems#

An allocation problem consists of player \(i = 1, \ldots, n\), a set of possible outcomes or allocations \(X\), (possibly) payments by each player \(p_1, \ldots, p_n\), and value functions \(v_i(x)\) such that the payoff for each player is \(v_i(x) - p_i\).
Challenge: the *mechanism designer* typically does not know the payoff functions! The function is the player’s privately known “type.”

#### “Good” Allocations#

Various criteria may define “good” allocations or outcomes. In auction problems, a good allocation might be one that is *Pareto efficient* given everyone’s values, or one that has high revenue for the seller. In matching problems, a good allocation might be one that is *stable*.

### Mechanisms#

- mechanism#
A mechanism \((M, x, p)\) is a triple consisting of:

A set of possible

*messages*for each player: \(M_1, \ldots, M_n\)An allocation rule: \(x(m_1, \ldots, m_n)\)

A payment rule \(p: M_1 \times \ldots \times M_n \to \mathbb{R}^n\) where player \(i\) pays \(p_i(m_1, \ldots, m_n)\).

We will look at mechanisms with and without payments. If there are no payments, set all \(p_i \equiv 0\).

- matching mechanism#
A

**matching mechanism**is a mapping from**reported**preferences into a matching. (no payments)- truthful mechanism#
A mechanism is

**truthful**if truthful reporting is a dominant action for each participant.- strategy-proof mechanism#
A

*strategy/action*for player \(i\) specifies what message to send as a function of player \(i\)’s type. A mechanism is*strategy-proof*if each player \(i\) has a dominant strategy, that is, a strategy that is optimal regardless of the strategies chosen by the other players. Sometimes, we’ll look at mechanisms that are not strategy-proof, using Nash equilibrium to forecast mechanism outcomes.

#### Example 1: Gayle-Shapley Marriage Problem#

In the Gayle-Shapley marriage problem, the messages are statements of preference (not necessarily truthful). The outcome/allocation/assignment is a matching, which the mechanism selects as a function of the reported preferences. No payments are made. A “type” is the preference list of each agent.

#### Example 2: Simple Auction#

In a simple auction problem, the messages are the bids \((b_i \geq 0)\). The auction mechanism determines who wins and what everyone must pay as a function of the bids. A type is typically a person’s value (maximum willingness to pay).

A mechanism plus payoffs defines a game. The possible actions are the messages: \(M_1, \ldots, M_n\). The payoffs are functions of the messages \(u_i(m_1, \ldots, m_n) = v_{i}\left(x\left(m_{1},\ldots,m_{n}\right)\right) - p_i(m_1, \ldots, m_n).\)

Often, we will focus on *direct mechanisms*: \(M_i\) is equal to the set of \(i\)’s possible payoff functions (or preferences).

### Applying GT to Market Design#

#### Studying existing markets#

Identify the “rules of the game,” the incentives for participants, and how they behave. Then try to understand why the market functions well, or not so well.

#### Designing new markets#

Identify the economic problem to be solved, the players and their incentives and information. Then try to understand what sort of market rules would lead to a good outcome when participants are selfish and smart.

Economic theory provides a conceptual framework, but good practice also uses data and experiment to test hypotheses and identify things models may have missed.

## Incentives in Matching#

Question

Can stable matching mechanisms be truthful?

### From Algorithms to Mechanisms#

Algorithm analysis assumes that inputs are given and correct. Mechanism analysis also examines the inputs and asks:

Is the mechanism truthful?

Is it easy for participants to know what to report?

(Man-Proposing Mechanism is Truthful for Men)

The direct mechanism that selects the man-optimal stable allocation using reported values (for example by running the man-proposing deferred acceptance algorithm) is truthful for the men.

Proof. Suppose man \(m\) makes report \(r\) that (fixing others’ reports) leads to matching \(x\) in which \(m\) is matched to woman \(w\): \(\mu_x(m) = w\). If instead of reporting \(r\), \(m\) reports that \(w\) is his only acceptable woman, \(x\) is still an acceptable stable matching to all parties. By the rural hospitals theorem, \(m\) is matched in every stable matching with the revised report. Since \(w\) is the only acceptable woman, DAA matches \(m\) to \(w\).

A key observation is that since \(m\) is matched in every stable matching, any acceptable matching \(y\) in which \(m\) is unmatched must be blocked by either \((m, w)\) or some \((m', w')\) with \(m \neq m'\).

Suppose that instead of reporting that only \(w\) is acceptable, \(m\) makes a “truncated truthful report” meaning that the women are ordered truthfully but those ranked below \(w\) are reported to be unacceptable. As before, \(m\) is matched in every stable matching even with the truncated report. Say the DAA now matches \(m\) to woman \(w'\) where \(w' \succeq_m w\). Note that under this truncated report, man \(m\) cannot be matched with a woman he likes less than woman \(w\).

Say that \(m\) had just reported truthfully, then the DAA would select the man-optimal stable matching \(\hat{x}\). Any matching that was blocked or unacceptable with the truncated report is still blocked or unacceptable in the same way. Hence \(\hat{x}\) is still the man-optimal stable matching and \(\mu_{\hat{x}}(m) = w'\).

Putting it all together, we get that for *any* reports by the women and the other men, if man \(m\) reports truthfully instead of reporting \(r\), he will be matched to woman \(w'\) instead of to woman \(w\) where \(w' \succ_m w\). \(\blacksquare\)

Exercise

Which steps of the proof also apply to women (in the man-offering deferred acceptance algorithm)?

What can you infer about the structure of women’s optimal reports when they play strategically?