1.2 — Essential Micro Concepts

# 1.2 — Essential Micro Concepts
## ECON 316 • Game Theory • Fall 2021
### Ryan Safner Assistant Professor of Economics <a href="mailto:safner@hood.edu">safner@hood.edu</a> <a href="https://github.com/ryansafner/gameF21">ryansafner/gameF21</a> <a href="https://gameF21.classes.ryansafner.com"> gameF21.classes.ryansafner.com</a>

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# Outline

### [Game Theory vs. Decision Theory](#3)
### [Optimization & Preferences](#)
### [Solution Concepts: Nash Equilibrium](#)

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# Game Theory vs. Decision Theory

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# The Two Major Models of Economics as a “Science”

## Optimization

- Agents have .hi[objectives] they value

- Agents face .hi[constraints]

- Make .hi[tradeoffs] to maximize objectives within constraints

## Equilibrium

- Agents .hi[compete] with others over **scarce** resources

- Agents .hi[adjust] behaviors based on prices

- .hi[Stable outcomes] when adjustments stop

]

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# Game Theory vs. Decision Theory Models I

- Traditional economic models are often called .hi[“Decision theory”]:

- .hi-purple[Optimization models] **ignore all other agents** and just focus on how can **you** maximize **your** objective within **your** constraints
  - Consumers max utility; firms max profit, etc.

- **Outcome**: .hi-purple[optimum]: decision where *you* have no better alternatives

]

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# Game Theory vs. Decision Theory Models I

- Traditional economic models are often called .hi[“Decision theory”]:

- .hi-purple[Equilibrium models] assume that there are **so many agents** that **no agent’s decision can affect the outcome**
  - Firms are price-takers or the *only* buyer or seller
  - **Ignores all other agents’ decisions**!

- **Outcome**: .hi-purple[equilibrium]: where *nobody* has any better alternative

]

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# Game Theory vs. Decision Theory Models III

- .hi[Game theory models] directly confront .hi-purple[strategic interactions] between players
  - How each player would optimally respond to a strategy chosen by other player(s)
  - Lead to a stable outcome where everyone has considered and chosen mutual best responses

- **Outcome**: .hi-purple[Nash equilibrium]: where *nobody* has a better strategy **given the strategies everyone else is playing**

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# Equilibrium in Games

- .hi-purple[Nash Equilibrium]: 
  - no player wants to change their strategy **given all other players’ strategies**
  - each player is playing a **best response** against other players’ strategies

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# Optimization & Preferences

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# Individual Objectives and Preferences

- What is a player's .hi-turquoise[objective] in a game? 
  - “To win”?
  - Few games are purely zero-sum
  
- “De gustibus non est disputandum”

- We need to know a player's .hi[preferences] over game outcomes

]

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# Modeling Individual Choice

- The .hi[consumer's utility maximization problem]:

1. **Choose:** .hi-purple[ < a consumption bundle >]

2. **In order to maximize:** .hi-green[< utility >]

3. **Subject to:** .hi-red[< income and market prices >]

]

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# Modeling Firm's Choice

1. **Choose:** .hi-blue[ < output >]

2. **In order to maximize:** .hi-green[< profits >]

- 2nd Stage: .hi-purple[firm's cost minimization problem]:

1. **Choose:** .hi-blue[ < inputs >]

2. **In order to _minimize_:** .hi-green[< cost >]

3. **Subject to:** .hi-red[< producing the optimal output >]
]

]

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# Preferences I

- Which game outcomes are **preferred** over others?

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**:] Between any two outcomes `$(a,b)$`:

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# Preferences II
.pull-left[

- We will allow **three possible answers**:

]

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# Preferences II

- We will allow **three possible answers**:

1. .blue[`\$a \succ b\$`: (Strictly) prefer `\$a\$` over `\$b\$`]
]

]

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# Preferences II

- We will allow **three possible answers**:

1. .blue[`\$a \succ b\$`: (Strictly) prefer `\$a\$` over `\$b\$`]

2. .blue[`\$a \prec b\$`: (Strictly) prefer `\$b\$` over `\$a\$`]

]

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# Preferences II

- We will allow **three possible answers**:

1. .blue[`\$a \succ b\$`: (Strictly) prefer `\$a\$` over `\$b\$`]

2. .blue[`\$a \prec b\$`: (Strictly) prefer `\$b\$` over `\$a\$`]

3. .blue[`\$a \sim b\$`: Indifferent between `\$a\$` and `\$b\$`]
]
]

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# Preferences II

- We will allow **three possible answers**:

1. .blue[`\$a \succ b\$`: (Strictly) prefer `\$a\$` over `\$b\$`]

2. .blue[`\$a \prec b\$`: (Strictly) prefer `\$b\$` over `\$a\$`]

3. .blue[`\$a \sim b\$`: Indifferent between `\$a\$` and `\$b\$`]
]

- .hi[*Preferences*] **are a list of all such comparisons between all bundles**

See my ECON 306 [class on preferences](https://micros21.classes.ryansafner.com/content/1.3-content/#appendix-1-material-on-preferences) for more.
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# So What About the Numbers?

.pull-left[
- Long ago (1890s), utility considered a real, measurable, .hi-purple[cardinal] scale.hi[†]

- Utility thought to be lurking in people's brains
  - Could be understood from first principles: calories, water, warmth, etc
    
- Obvious problems
]

.footnote[.hi[†] "Neuroeconomics" & cognitive scientists are re-attempting a scientific approach to measure utility]

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# Utility Functions?

- More plausibly .hi-turquoise[infer people's preferences from their actions]!
  - “Actions speak louder than words”

- .hi-purple[Principle of Revealed Preference]: if a person chooses `$x$` over `$y$`, and both are affordable, then they must prefer `$x \succeq y$`
	
- Flawless? Of course not. But extremely useful approximation!
  - People tend not to leave money on the table
]

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# Utility Functions!

- A .hi[utility function] `$u(\cdot)$`.hi[†] *represents* preference relations `$(\succ , \prec , \sim)$`

- Assign utility numbers to bundles, such that, for any bundles `$a$` and `$b$`:
`$$a \succ b \iff u(a)>u(b)$$`
]

.footnote[.hi[†] The `\$\cdot\$` is a placeholder for whatever goods we are considering (e.g. `\$x\$`, `\$y\$`, burritos, lattes, dollars, etc)]

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# Utility Functions, Pural I

.pull-left[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.hi-green[Example]: Imagine three alternative bundles of `$(x, y)$`:
`$$\begin{aligned}
a&=(1,2)\\
b&=(2,2)\\
c&=(4,3)\\
\end{aligned}$$`
]
]

| `$u(\cdot)$` |
|------------|
| `$u(a)=1$`   | 
| `$u(b)=2$`   |
| `$u(c)=3$`   |

]

- .hi-turquoise[Does this mean that bundle `\$c\$` is 3 times the utility of `\$a\$`?]

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# Utility Functions, Pural II

| `$u(\cdot)$` | `$v(\cdot)$` |
|------------|------------|
| `$u(a)=1$`   | `$v(a)=3$`   |
| `$u(b)=2$`   | `$v(b)=5$`   |
| `$u(c)=3$`   | `$v(c)=7$`   |

]

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# Utility Functions, Pural III

- Both are valid utility functions:
    - `$u(c)>u(b)>u(a)$` ✅
    - `$v(c)>v(b)>v(a)$` ✅
    - because `$c \succ b \succ a$`

- .hi-purple[Only the .ul[ranking] of utility numbers matters!]

]

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