3.1 — Mixed Strategies

# 3.1 — Mixed Strategies
## ECON 316 • Game Theory • Fall 2021
### Ryan Safner<br> Assistant Professor of Economics <br> <a href="mailto:safner@hood.edu"><i class="fa fa-paper-plane fa-fw"></i>safner@hood.edu</a> <br> <a href="https://github.com/ryansafner/gameF21"><i class="fa fa-github fa-fw"></i>ryansafner/gameF21</a><br> <a href="https://gameF21.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>gameF21.classes.ryansafner.com</a><br>

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# Outline

### [When Pure Strategies Won't Work](#3)
### [MSNE in Constant Sum Games](#16)
### [Coordination Games: PSNE and MSNE](#67)

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# When Pure Strategies Won't Work

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# When Pure Strategies Won't Work

.center[
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]

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# When Pure Strategies Won't Work

1902—1977
]]
]

.right-column[
.smallest[
> “Sherlock Holmes, pursued by his opponent, Moriarty, leaves London for Dover. The train stops at a station on the way, and he alights there rather than travelling on to Dover. He has seen Moriarty at the railway station, recognizes that he is very clever and expects that Moriarty will take a faster special train in order to catch him in Dover. Holmes’s anticipation turns out to be correct. But what if Moriarty had been still more clever, had estimated Holmes’s mental abilities better and had foreseen his actions accordingly? Then, obviously, he would have travelled to the intermediate station [Canterbury]. Holmes again would have had to calculate that, and he himself would have decided to go on to Dover. Whereupon, Moriarty would again have ‘reacted’ differently,” (p.173-4).

]

---

# When Pure Strategies Won't Work

> “‘All that I have to say has already crossed your mind,’ said he.
> ‘Then possibly my answer has crossed yours,’ I replied.
> ‘You stand fast?’
> ‘Absolutely.’”

— Arthur Conan Doyle, 1893, *The Final Problem*
]

---

# When Pure Strategies Won't Work

.center[
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]

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# When Pure Strategies Won't Work

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# Expected Value

- .hi[Expected value] of a random variable `$X$`, written `$E(X)$` (and sometimes `$\mu)$`, is the long-run average value of `$X$` "expected" after many repetitions

`$$E(X)=\sum^k_{i=1} p_i x_i$$`

- `$E(X)=p_1x_1+p_2x_2+ \cdots +p_kx_k$`

- A **probability-weighted average** of `$X$`, with each possible `$X$` value, `$x_i$`, weighted by its associated probability `$p_i$`

- Also called the .hi["mean"] or .hi["expectation"] of `$X$`, always denoted either `$E(X)$` or `$\mu_X$`

---

# Expected Value: Example I

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Suppose you lend your friend $100 at 10% interest. If the loan is repaid, you receive $110. You estimate that your friend is 99% likely to repay, but there is a default risk of 1% where you get nothing. What is the expected value of repayment?

]

---

# Mixed Strategies

- .hi[Pure strategy]: is a complete strategy profile that a player will play
  - Recall, .hi[strategy] is a list of choices player will take at every possible decision node
  
- .hi[Mixed strategy] is a **probability distribution** over a strategy profile
  - Plays a variety of pure strategies according to probabilities
]

---

# Mixed Strategies

- The logic of mixed strategies is best understood in the context of repeated constant-sum games

- If you play one strategy repeatedly (i.e. a .hi[pure strategy]), your opponent can exploit your (predictable) strategy with their best response

- You want to “keep your opponent guessing”
]

---

# Mixed Strategy Nash Equilibrium

- We have already seen Nash equilibrium in pure strategies (PSNE)

- Nash (1950) proved that any `$n$`-player game with a finite number of pure strategies has at least one equilibrium
  - A game may have no PSNE, but there will always be a unique .hi[mixed strategy Nash equilibrium (MSNE)]
  - Games may have *both* pure and a mixed NE
]

---

# Mixed Strategy Nash Equilibrium

.pull-left[
.smallest[
- Finding this is relatively straightforward with two players and two strategies

1. Let *p* be the probability of one player playing one of their available strategies
  - Let *(1-p)* be the probability of that player playing their other available strategy
2. Let *q* be the probability of the other player playing one of their available strategies
  - Let *(1-q)* be the probability of that player playing their other available strategy

- .hi-turquoise[There exists some *(p,q)* mix that is a Nash equilibrium in mixed strategies]
]
]
.pull-right[
.center[
![](../images/coin-flip.png)
]
]

---

# MSNE in Constant Sum Games

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# MSNE in Constant Sum Games

.pull-left[
- Consider the following game between a .hi-red[Kicker] and a .hi-blue[Goalie] during a penalty kick
]

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# MSNE in Constant Sum Games

.pull-left[
- Consider the following game between a .hi-red[Kicker] and a .hi-blue[Goalie] during a penalty kick

- A constant sum game (in this case, zero-sum)
  - If both choose same direction, .hi-blue[Goalie] blocks goal
  - If both choose different directions, .hi-red[Kicker] gets goal
]

---

# MSNE in Constant Sum Games

.pull-left[
- Palacios-Huerta (2003) calculated average success rates in English, Spanish, & Italian leagues (1995-2000)

- If both .hi-red[Kicker] and .hi-blue[Goalie] choose same direction, .hi-red[Kicker]'s payoff is higher if he chooses his natural side (often .red[Right])

]

.source[Palacios-Huerta, Ignacio, 2003, “Professionals Play Minimax,” *Review of Economic Studies* 70(2): 395–415]

---

# MSNE in Constant Sum Games

.pull-left[
- This game has no Nash equilibrium in pure strategies (PSNE)
  - From any outcome, at least one player would prefer to switch strategies
  - No outcome has *all* players playing a best response

]

---

# MSNE in Constant Sum Games

.pull-left[
- What if .hi-red[Kicker] were to .hi-purple[randomize] strategies
  - Say 50% of the time, .red[Kick Left], 50% of the time, .red[Kick Right]

- Let `$p$` be probability that .hi-red[Kicker] plays .red[Kick Left]
  - `$p = 0.50$`
]

---

# MSNE in Constant Sum Games

.pull-left[
- Then .hi-blue[Goalie] wants to maximize his .hi-purple[expected] payoff, given .hi-red[Kicker] plays .red[Kick Left] with `$p=0.50$`

]

---

# MSNE in Constant Sum Games

.pull-left[
- Then .hi-blue[Goalie] wants to maximize his .hi-purple[expected] payoff, given .hi-red[Kicker] plays .red[Kick Left] with `$p=0.50$`

- If .hi-blue[Goalie] plays .blue[Dive Left]:
`$$\begin{align*}\mathbb{E}[\color{blue}{\text{Dive Left}}] &= \color{blue}{42}(p)+\color{blue}{7}(1-p)\\
&= \color{blue}{42}(0.50)+\color{blue}{7}(1-0.50)\\
\end{align*}$$`
  - He can .hi-purple[expect] to earn .blue[24.5]
]

---

# MSNE in Constant Sum Games

.pull-left[
- Then .hi-blue[Goalie] wants to maximize his .hi-purple[expected] payoff, given .hi-red[Kicker] plays .red[Kick Left] with `$p=0.50$`

- If .hi-blue[Goalie] plays .blue[Dive Right]:
`$$\begin{align*}\mathbb{E}[\color{blue}{\text{Dive Right}}] &= \color{blue}{5}(p)+\color{blue}{30}(1-p)\\
&= \color{blue}{5}(0.50)+\color{blue}{30}(1-0.50)\\
\end{align*}$$`
  - He can .hi-purple[expect] to earn .blue[17.5]
]

---

# MSNE in Constant Sum Games

.pull-left[
- Then .hi-blue[Goalie] wants to maximize his .hi-purple[expected] payoff, given .hi-red[Kicker] plays .red[Kick Left] with `$p=0.50$`

- .hi-blue[Goalie] will play .blue[Dive Left] to maximize his expected payoff (.blue[24.5] `$\succ$` .blue[17.5])

]

---

# MSNE in Constant Sum Games

- Since .hi-blue[Goalie] will .blue[Dive Left] to maximize his expected payoff, .hi-red[Kicker] can expect to earn:

`$$\begin{align*}
&\color{red}{58}(p)+\color{red}{93}(1-p)\\
&\color{red}{58}(0.50)+\color{red}{93}(1-0.50)\\
&\color{red}{75.5}
\\
\end{align*}$$`

- .hi-blue[Goalie] playing .blue[Dive Left] holds .hi-red[Kicker]'s expected payoff down to 75.5
]

---

# The Minimax Theorem

- In constant sum games, note that even in *mixed* strategies, one player increases their own (expected) payoff by pulling down the other player's (expected) payoff!

- In this game, even expected payoffs always sum to 100
  - .hi-red[Kicker]'s `$\mathbb{E[\pi]}=75.5$`
  - .hi-red[Goalie]'s `$\mathbb{E[\pi]}=24.5$`
]

# The Minimax Theorem

- von Neumann & Morgenstern’s .hi[minimax theorem] (simplified): in a 2-person, constant sum game, each player maximizes their own expected payoff by minimizing their opponent's expected payoff

- The name .hi[“minimax”] is a popular strategy in games, trying to minimize the risk of your maximum possible loss
]

---

# Penalty Kicks: 50:50?

- .hi-red[Kicker]'s “randomizing” 50:50 (.red[Kick Left], .red[Kick Right]) was not random enough!

- .hi-blue[Goalie] recognizing this pattern can exploit it and hold down .hi-red[Kicker]'s expected payoff

- .hi-red[Kicker] can do better by picking a better `$p$` (and similarly, so can .hi-blue[Goalie])
  - Hint: if .hi-blue[Goalie] knew .hi-red[Kicker]'s `$p$` before .hi-blue[Goalie] chose, would he have a clearly better choice of .blue[Dive Left] vs. .blue[Dive Right]?
]

---

# The Opponent Indifference Principle

.smallest[
- Want to find the optimal probability mix that .hi-purple[leaves your opponent(s) _indifferent_ between their strategies to respond]

- In constant sum games (i.e. sports, board games, etc)
  - Making your opponent indifferent `$\implies$` minimizing your opponent's ability to recognize & exploit patterns in your actions

- This principle is the same in non-constant sum games too!

- Implies game is played repeatedly

- Not always intuitive, but a simple principle
]
]
.pull-right[
.center[
![](../images/choices.jpg)
]
]

---

# Kicker's Optimal Choice of p

- We want to find .hi-red[Kicker]'s optimal mixed strategy that leaves .hi-blue[Goalie] indifferent between his (pure) strategies

- Suppose .hi-red[Kicker] plays .red[Kick Left] with probability .red[p]
]

---

# Kicker's Optimal Choice of p

- We want to find .hi-red[Kicker]'s optimal mixed strategy that leaves .hi-blue[Goalie] indifferent between his (pure) strategies

- Suppose .hi-red[Kicker] plays .red[Kick Left] with probability .red[p]

- .hi-blue[Goalie]'s expected payoff of playing .blue[Dive Left]: .blue[42]p+.blue[7](1-p)
]
]
.pull-right[
.center[
![](../images/penalty_kick_game_palacios.png)
]
]

---

# Kicker's Optimal Choice of p

.pull-left[
.smallest[
- We want to find .hi-red[Kicker]'s optimal mixed strategy that leaves .hi-blue[Goalie] indifferent between his (pure) strategies

- Suppose .hi-red[Kicker] plays .red[Kick Left] with probability .red[p]

- .hi-blue[Goalie]'s expected payoff of playing .blue[Dive Left]: .blue[42]p+.blue[7](1-p)

- .hi-blue[Goalie]'s expected payoff of playing .blue[Dive Right]: .blue[5]p+.blue[30](1-p)
]
]

---

# Kicker's Optimal Choice of p

.pull-left[
.smallest[
- We want to find .hi-red[Kicker]'s optimal mixed strategy that leaves .hi-blue[Goalie] indifferent between his (pure) strategies

- Suppose .hi-red[Kicker] plays .red[Kick Left] with probability .red[p]

- .hi-blue[Goalie]'s expected payoff of playing .blue[Dive Left]: .blue[42]p+.blue[7](1-p)

- .hi-blue[Goalie]'s expected payoff of playing .blue[Dive Right]: .blue[5]p+.blue[30](1-p)

- What value of .red[p] would make .hi-blue[Goalie] indifferent between .blue[Dive Left] and .blue[Dive Right]?
  - i.e. `$\mathbb{E}[\color{blue}{Left}] = \mathbb{E}[\color{blue}{Right}]$`
]
]
.pull-right[
.center[
![](../images/penalty_kicks_game_p.png)
]
]

---

# Kicker's Optimal Choice of p, Graphically

---

# Kicker's Optimal Choice of p: Algebraically

- Find value of .red[p] that equates .hi-blue[Goalie]'s expected payoff of .blue[Dive Left] and .blue[Dive Right]:

`$$\begin{align*}
\mathbb{E}[\color{blue}{Left}] &= \mathbb{E}[\color{blue}{Right}] \\
\mathbb{E}[\color{blue}{42}p+\color{blue}{7}(1-p)] &= \mathbb{E}[\color{blue}{5}p+\color{blue}{30}(1-p)] \\
\end{align*}$$`
--

- `$p^\star = 0.383$`

- .hi-red[Kicker] plays .red[Kick Left] with `$p=0.383$` and .red[Kick Right] with `$1-p = 0.617$`

- .hi-blue[Goalie]'s expected payoff of .blue[Dive Left]: `$\color{blue}{42}(0.383)+\color{blue}{7}(0.617) \approx \color{blue}{20.41}$`

- .hi-blue[Goalie]'s expected payoff of .blue[Dive Right]: `$\color{blue}{5}(0.383)+\color{blue}{30}(0.617) \approx \color{blue}{20.41}$`

---

# Goalie's Optimal Choice of q

.pull-left[
.smallest[
- We want to find .hi-blue[Goalie]'s optimal mixed strategy that leaves .hi-red[Kicker] indifferent between his (pure) strategies

- Suppose .hi-blue[Goalie] plays .blue[Dive Left] with probability .blue[q]
]
]

---

# Goalie's Optimal Choice of q

.pull-left[
.smallest[
- We want to find .hi-blue[Goalie]'s optimal mixed strategy that leaves .hi-red[Kicker] indifferent between his (pure) strategies

- Suppose .hi-blue[Goalie] plays .blue[Dive Left] with probability .blue[q]

- .hi-red[Kicker]'s expected payoff of playing .red[Dive Left]: .red[58]q+.red[95](1-q)
]
]

---

# Goalie's Optimal Choice of q

.pull-left[
.smallest[
- We want to find .hi-blue[Goalie]'s optimal mixed strategy that leaves .hi-red[Kicker] indifferent between his (pure) strategies

- Suppose .hi-blue[Goalie] plays .blue[Dive Left] with probability .blue[q]

- .hi-red[Kicker]'s expected payoff of playing .red[Dive Left]: .red[58]q+.red[95](1-q)

- .hi-red[Kicker]'s expected payoff of playing .blue[Dive Right]: .red[93]q+.red[70](1-q)

]
]

---

# Goalie's Optimal Choice of q

.pull-left[
.smallest[
- We want to find .hi-blue[Goalie]'s optimal mixed strategy that leaves .hi-red[Kicker] indifferent between his (pure) strategies

- Suppose .hi-blue[Goalie] plays .blue[Dive Left] with probability .blue[q]

- .hi-red[Kicker]'s expected payoff of playing .red[Dive Left]: .red[58]q+.red[95](1-q)

- .hi-red[Kicker]'s expected payoff of playing .blue[Dive Right]: .red[93]q+.red[70](1-q)

- What value of .red[p] would make .hi-red[Kicker] indifferent between .red[Kick Left] and .red[Kick Right]?
  - i.e. `$\mathbb{E}[\color{red}{Left}] = \mathbb{E}[\color{red}{Right}]$`

]
]

---

# Goalies's Optimal Choice of q, Graphically

---

# Goalie's Optimal Choice of q: Algebraically

- Find value of .blue[q] that equates .hi-red[Kicker]'s expected payoff of .red[Kick Left] and .red[Kick Right]:

`$$\begin{align*}
\mathbb{E}[\color{red}{Left}] &= \mathbb{E}[\color{red}{Right}] \\
\mathbb{E}[\color{red}{58}q+\color{red}{95}(1-q)] &= \mathbb{E}[\color{red}{93}q+\color{red}{70}(1-q)] \\
\end{align*}$$`

- `$q^\star = 0.417$`

- .hi-blue[Goalie] plays .blue[Dive Left] with `$q=0.417$` and .blue[Dive Right] with `$1-q = 0.583$`

- .hi-red[Kicker]'s expected payoff of .red[Kick Left]: `$\color{red}{58}(0.417)+\color{red}{95}(0.583) \approx \color{red}{79.57}$`
  
--

- .hi-red[Kicker]'s expected payoff of .red[Kick Right]: `$\color{red}{93}(0.417)+\color{red}{70}(0.583) \approx \color{red}{79.57}$`

---

# Mixed Strategy Nash Equilibrium

- .hi-blue[Goalie] is indifferent between .blue[Dive Left] and .blue[Dive Right] when .hi-red[Kicker] plays .red[Kick Left] with .red[p]=0.383

- .hi-red[Kicker] is indifferent between .red[Kick Left] and .red[Kick Right] when .hi-blue[Goalie] plays .blue[Dive Left] with .blue[q]=0.417

]

---

# Mixed Strategy Nash Equilibrium

- .hi-blue[Goalie] is indifferent between .blue[Dive Left] and .blue[Dive Right] when .hi-red[Kicker] plays .red[Kick Left] with .red[p]=0.383

- .hi-red[Kicker] is indifferent between .red[Kick Left] and .red[Kick Right] when .hi-blue[Goalie] plays .blue[Dive Left] with .blue[q]=0.417

- .hi-purple[Mixed Strategy Nash Equilibrium (MSNE)]: (.red[p], .blue[q]) = (.red[0.383], .blue[0.417])
]

---

# Mixed Strategy Nash Equilibrium

- .hi-blue[Goalie] is indifferent between .blue[Dive Left] and .blue[Dive Right] when .hi-red[Kicker] plays .red[Kick Left] with .red[p]=0.383

- .hi-red[Kicker] is indifferent between .red[Kick Left] and .red[Kick Right] when .hi-blue[Goalie] plays .blue[Dive Left] with .blue[q]=0.417

- .hi-purple[Mixed Strategy Nash Equilibrium (MSNE)]: (.red[p], .blue[q]) = (.red[0.383], .blue[0.417])
  - .hi-red[Kicker]'s expected payoff: .red[79.57] 
  - .hi-blue[Goalie]'s expected payoff: .blue[20.41]
  - Note they sum to 1!

]

---

# p and q as Best Responses

]
]

---

# p and q as Best Responses

- .hi-blue[Goalie].blue['s Best Response] `$=\begin{cases} Right & \text{if }p<0.383\\ Indifferent & \text{if }p=0.383\\ Left & \text{if }p>0.383\\ \end{cases}$` 
]
]

---

# p and q as Best Responses

- .hi-blue[Goalie].blue['s Best Response] `$=\begin{cases} Right & \text{if }p<0.383\\ Indifferent & \text{if }p=0.383\\ Left & \text{if }p>0.383\\ \end{cases}$`

- .hi-red[Kicker].red['s Best Response] `$=\begin{cases} Left & \text{if }q<0.417\\ Indifferent & \text{if }q=0.417\\ Right & \text{if }q>0.417\\ \end{cases}$`
]
]

---

# p and q as Best Responses

.pull-left[
.smallest[
- .red[p] = pr(.red[Kicker kicks Left])
- .blue[q] = pr(.blue[Goalie dives Left])

- .hi-blue[Goalie].blue['s Best Response] `$=\begin{cases} Right & \text{if }p<0.383\\ Indifferent & \text{if }p=0.383\\ Left & \text{if }p>0.383\\ \end{cases}$`

- .hi-red[Kicker].red['s Best Response] `$=\begin{cases} Left & \text{if }q<0.417\\ Indifferent & \text{if }q=0.4173\\ Right & \text{if }q>0.417\\ \end{cases}$`

- Like any Nash equilibrium, players are playing mutual best responses to each other (probabilistically)
]
]
.pull-right[
.center[
![](../images/penalty_kicks_game_msne.png)
]
]

---

# Goalie's Best Reponse (q) to p

.hi-blue[Goalie].blue['s Best Response] `$=\begin{cases} Right & \text{if }p<0.383\\ Indifferent & \text{if }p=0.383\\ Left & \text{if }p>0.383\\ \end{cases}$`

]

.pull-right[
<img src="3.1-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Kicker's Best Reponse (p) to q

- .hi-red[Kicker].red['s Best Response] `$=\begin{cases} Left & \text{if }q<0.417\\ Indifferent & \text{if }q=0.4173\\ Right & \text{if }q>0.417\\ \end{cases}$`

]

.pull-right[
<img src="3.1-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Mixed Strategy Nash Equilibrium

- Like any Nash equilibrium, where best response functions intersect

]
.pull-right[
<img src="3.1-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Rock-Paper-Scissors I

- A two player game with *three* strategies available to each

- Graphically more difficult, but same principle to find .hi[MSNE]
  - find probabilities that make opponent indifferent between their responses

- Game is symmetric, so only need to find one player's optimal mixed strategy

]

---

# Rock-Paper-Scissors II

- Define for .hi-blue[Column]:
  - `$r =$` pr(.blue[Rock])
  - `$p =$` pr(.blue[Paper])
  - `$1-r-p$` = pr(.blue[Scissors])
]

---

# Rock-Paper-Scissors II

- Define for .hi-blue[Column]:
  - `$r =$` pr(.blue[Rock])
  - `$p =$` pr(.blue[Paper])
  - `$1-r-p$` = pr(.blue[Scissors])

- .hi-blue[Column] must choose `$r, p$` that make .hi-red[Row] indifferent between their strategies
]

---

# Rock-Paper-Scissors II

- List the expected payoffs to .hi-red[Row] from .hi-blue[Column]'s mix of `$r,p$`

- .hi-red[Row]'s expected payoff must equal for all three strategies
  - So let's take any two and set them equal:
  
`$$\begin{align*}
2r+p-1 &= p-r
\end{align*}$$`
]

---

# Rock-Paper-Scissors II

- List the expected payoffs to .hi-red[Row] from .hi-blue[Column]'s mix of `$r,p$`

- .hi-red[Row]'s expected payoff must equal for all three strategies
  - So let's take any two and set them equal:
  
`$$\begin{align*}
2r+p-1 &= p-r
\end{align*}$$`

- `$r = \frac{1}{3}$`
- `$p = \frac{1}{3}$`
- `$(1-r-p) = \frac{1}{3}$`
]

---

# Rock-Paper-Scissors II

- .hi-purple[MSNE]: each player plays all three strategies with equal probability `$\left(\frac{1}{3}\right)$`
]

---

# Coordination Games: PSNE and MSNE

---

# MSNE in Coordination Games

- The necessity of MSNE is easy to see for constant-sum games with no PSNE

- But MSNE also exist for non-constant sum games, and for games with one or more PSNE
]

---

# Assurance Game: MSNE

- We know an .hi-purple[assurance game] has two PSNE

- Let's solve for MSNE
]

---

# Assurance Game: MSNE

- Let `$p=$`pr(.hi-red[Harry] goes to .red[Whitaker])

- Let `$q=$`pr(.hi-blue[Sally] goes to .blue[Whitaker])

]

---

# Assurance Game: MSNE

- Let `$p=$`pr(.hi-red[Harry] goes to .red[Whitaker])

- Let `$q=$`pr(.hi-blue[Sally] goes to .blue[Whitaker])

]

---

# Assurance Game: MSNE

- Let `$p=$`pr(.hi-red[Harry] goes to .red[Whitaker])

- Let `$q=$`pr(.hi-blue[Sally] goes to .blue[Whitaker])

- `$p^\star = \frac{1}{3}$`
- `$q^\star = \frac{1}{3}$`

- .hi-purple[MSNE]: (.red[p], .blue[q]) = `$(\color{red}{\frac{1}{3}}, \color{blue}{\frac{1}{3}})$`
]

---

# Assurance Game: MSNE

- Calculate expected payoffs to .hi-red[Harry] and .hi-blue[Sally] with (.red[p], .blue[q]) MSNE

]

---

# Assurance Game: MSNE

- Calculate expected payoffs to .hi-red[Harry] and .hi-blue[Sally] with (.red[p], .blue[q]) MSNE
  - .hi-red[Harry]: `$\color{red}{\frac{2}{3}}$`
  - .hi-blue[Sally]: `$\color{blue}{\frac{2}{3}}$`

]

---

# Assurance Game: MSNE

- Problem: MSNE is even worse than either PSNE in this game!
  - Significant probability of going to different places
  - Also very fragile, anything `$>,<\frac{1}{3}$` reverts to PSNE
]

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# Assurance Game: MSNE

- .hi-blue[Sally's BR] `$=\begin{cases} Starbucks & \text{if }p<\frac{1}{3}\\ Indifferent & \text{if }p=\frac{1}{3}\\ Whitaker & \text{if }p>\frac{1}{3}\\ \end{cases}$`

- .hi-red[Harry's BR] `$=\begin{cases} Starbucks & \text{if }q<\frac{1}{3}\\ Indifferent & \text{if }q=\frac{1}{3}\\ Whitaker & \text{if }q>\frac{1}{3}\\ \end{cases}$`
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# Assurance Game: MSNE

- All intersections of best response functions are Nash equilibria

- Interior solution: MSNE

- Corner solutions: PSNE
  - PSNE are special cases of MSNE where `$p \in \{0,1\}$` and `$q \in \{0,1\}$`
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