2. Normal-form games

2.4 Nash Equilibrium (Chapter 9)

Examples: Location, Partnership Games (Chapter 8)

Recall that ISD is driven only by common knowledge of rationality

However, ISD makes a weak prediction in many games

In order to obtain a sharp prediction, therefore, we need impose restrictions on players’ **beliefs** about their opponents’ behavior. This will lead to the central solution concept of game theory, *Nash equilibrium*.

### Definition

DEFINITION 2.10. A strategy profile s* = \((s_{1}^{*}, s_{2}^{*})\) is a **Nash equilibrium** if for each player i, \(s_{i}^{*}\) is a best response to \(s_{j}^{*}\).

This implies,

\(u_{1}(s_{1}^{*}, s_{2}^{*})\) ≥ \(u_{1}(s_{1}, s_{2}^{*})\) for every \(s_{1} \in S_{1}\)

and

\(u_{2}(s_{1}^{*}, s_{2}^{*})\) ≥ \(u_{2}(s_{1}, s_{2}^{*})\) for every \(s_{2} \in S_{2}\)

Simply put, a Nash equilibrium is the players’ *mutual* best responses

Two underlying assumptions:

(i) Each player has *correct beliefs* about what opponents will do (equilibrium knowledge)

(ii) Each behaves *rationally* given these beliefs

The Nash equilibrium first appeared in 1952. If we think about an equilibrium where there is no risk of deviation from economics, the other player must know exactly what they are going to do.

But that’s too much of an assumption to make, so we start thinking about Rationality alone.

This is where ISD comes in, and then Rationalizability comes in because there’s not much we can do with ISD after that.

### Justifications of Equilibrium Knowledge

##### 1. Learning or evolution

- A game is repeatedly played in society or by a group of agents. The behavior of the players settles down in that the same strategies are used each time the game is played.

##### 2. Pre-play communication

- The players meet before playing a game and reach an agreement on the strategy that each will use. Subsequently, the players individually honor the agreement.

##### 3. Coordinatiton of play by a mediator (Contractual relationship, See Section 2.8)

- An outside mediator recommends to the players that they adopt a specific strategy. Each player, expecting that the others will follow the recommendation, has the right incentive to follow it as well.

##### 4. Focal points (THOMAS SCHELLING (1960))

- Something about the game makes Nash eqbm the obvious choice about how to behave.

### Computing Nash Equilibria

The next theorem provides a link between Nash eqbm and ISD

THEOREM 2.11. If a normal-form game is solvable by ISD, the strategy profile surviving the process constitutes a *unique* Nash equilibrium of the game.

Nash eqbm is much more assumption-based than ISD.

If ISD solves the problem, the remaining profile is the only Nash equilibrium in the game.

EXAMPLE 2.13 (REVISIT PRISONER’S DILEMMA).

1 \ 2 | C | D |

C | 2, 2 | 0, 3 |

D | 3, 0 | 1, 1 |

ISD leaves only one pure strategy profile, (D, D), which constitutes a unique Nash equilibrium of this game

Theorem 2.11 also provides guidelines to compute Nash eqbm

(1) Eliminate pure strategies that are strictly dominated. If this leaves only one pure strategy profile, then we are done: it becomes the unique Nash eqbm

(2) For each profile of supports, find all equilibria.

EXAMPLE 2.14. Find all pure-strategy Nash equilibria of the following game:

1 \ 2 | X | Y | Z |

J | 5, 6 | 3, 7 | 0, 4 |

K | 8, 3 | 3, 1 | 5, 2 |

L | 7, 5 | 4, 4 | 5, 6 |

M | 3, 4 | 7, 5 | 3, 3 |

strategy J is strictly dominated by L

If player 1 chooses K, then player 2’s best response is X

If player 1 chooses L, then player 2’s best response is Z

If player 1 chooses M, then player 2’s best response is Y

If player 2 chooses X, then player 1’s best response is K

If player 2 chooses Y, then player 1’s best response is M

If player 2 chooses Z, then player 1’s best response are K, L

So the nash equilibrium is (K, X), (M, Y), (L, Z).

EXAMPLE 2.15 (PARTNERSHIP GAME, CHAPTER 8). Consider a partnership between two players. The joint profit is given by

4(x + y + cxy).

where c ∈ [0, 1/4] measures the degree of **complementarity** in the partnership.

Suppose the cost of effort is \(x^{2}\) for player 1 and \(y^{2}\) for player 2.

player 1 is x, player 2 is y related to

complementarity, meaning how important the collaboration is

1) player: N = {1, 2}

2) strategy: \(S_{1}\) = [0, ∞), \(S_{2}\) = [0, ∞)

3) \(u_{1}\)(x, y) = 2(x + y + cxy) – \(x^{2}\)

\(u_{2}\)(x, y) = 2(x + y + cxy) – \(y^{2}\)

Need to divide joint profit in half and subtract cost of effort

These three elements are the normal-form representation (player, player strategy, payoff function)

player 1’s payoff function \(\frac{\partial u_{1}}{\partial x}\) = 2 + 2cy – 2x = 0 ⇔ x = 1 + cy ⇔ y = (x – 1)/c

player 2’s payoff function \(\frac{\partial u_{2}}{\partial y}\) = 2 + 2cx – 2y = 0 ⇔ y = 1 + cx

1+cx = (x-1)/c ⇔ c + \(c^{2}\)x = x-1 ⇔ (\(c^{2}\)-1)x = -c-1 ⇔ x = \((1-c)^{-1}\) = \(\frac{1}{1-c}\)

y = 1+cx = 1+c∙\(\frac{1}{1-c}\) = \(\frac{(1-c)+c}{1-c}\) = \(\frac{1}{1-c}\))

EXAMPLE 2.16 (LOCATION GAME, EXERCISE 9.4 ON PAGE 109). There are two players who simultaneously select numbers between 0 and 1. Suppose player 1 chooses \(s_{1}\) and player 2 chooses \(s_{2}\). For each profile \((s_{i}, s_{j})\), player i’s payoff is

\(u_{i}(s_{i}, s_{j})\) = \(\begin{cases} & \frac{s_{i}+s_{j}}{2} \text{ if } s_{i} < s_{j} \ & 1-\frac{s_{i}+s_{j}}{2} \text{ if } s_{i} > s_{j} \ & \frac{1}{2} \text{ if } s_{i} = s_{j} \end{cases}\)

Compute the Nash equilibria of this game.

Since the player’s payoff function is **discontinuous**, and thus standard technique does not work

Key observation: \(s_{i} < s_{j}\) cannot arise in any eqbm and thus \(s_{1} = s_{2}\) is the case

- \(s_{1} = s_{2} < \frac{1}{2}\) cannot be an eqbm, because each player has an incentive to deviate
- \(s_{1} = s_{2} > \frac{1}{2}\) cannot be an eqbm, because each player has an incentive to deviate

Hence the only candidate is

\(s_{1} = s_{2} = \frac{1}{2}\)

In political science, this model is used to explain optimal political positions of candidates (known as the *median voter theorem*)

- Reference1: Chang-Koo Chi, (7/50) Game Theory and Applications 2 – Nash Equilibrium, Jul 1, 2020, https://youtu.be/Z0DL4ss-Emw
- Reference2: Chang-Koo Chi, (8/50) Game Theory and Applications 2 – Location Game, Jul 1, 2020, https://youtu.be/Sbutb6fvSJ8