Change in the Method of Analysis
Illustrating Sim. Games using Trees
Martina Navratilova v. Chris Evert
| DL |
50, 50 |
80, 20 |
| CC |
90, 10 |
20, 80 |
Navratilova has volleyed the ball to Evert
Evert can send the ball down the line, DL, or crosscourt CC (a softer diagonal shot)
Navratilova has to simultaneously choose to set up to receive either type of shot
Illustrating Sim. Games using Trees
The simultaneous tennis game can also be represented as a tree
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The information set indicates that Navratilova doesn’t know which shot before she has to choose how to receive
Adding Nature as a player
Even in cases which aren’t actually strategic, we can still use some of these tools
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Changing the order of Moves
We found the NE will be (Deficit, High Rates) when this game is played simultaneously
| Balance |
3, 4 |
1, 3 |
| Deficit |
4, 1 |
2, 2 |
Changing the order of Moves
What about when the Fed moves first?
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Changing the order of Moves
What about when Congress moves first?
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now we get a different NE with congress playing their strictly dominated strategy
this is because when the Fed acts second, they can make a credible threat
SPNE
The rollback method finds the subgame-perfect Nash equilibria of sequential games
- A complete plan of action such that at every node including off-equilibrium paths, it is still rational to stick to that original plan of action.
Market Entry Game
Consider this variation on the Entry Game, in which the Entrant is not as efficient as the Incumbent:
| \(Enter\) |
-1, 1 |
1, 2 |
| \(Out\) |
0, 3 |
0, 3 |
- Note that this game has two NEs: (Enter, Accommodate) and (Out, Fight).
- Does it really make sense for the Incumbent to ever Fight?
Refining Nash Equilibrium
The more you think about it, the less sense this Nash equilibrium makes.
Entrant would stay Out if they really, truly believe that if they Entered, the Incumbent would Fight.
… but it’s hard to justify that belief when Fighting is costly for the Incumbent.
The Incumbent would have to somehow convince the Entrant that they really would Fight them,
- perhaps by making a credible threat.
We will talk more about this kind of strategic move later in the course.
We often don’t want this kind of Nash Equilibrium. To rule it out, we have the concept of subgame perfection.
Subgames
A subgame is a subset of an extensive-form game which, itself, is a complete game.
A subgame is a single node, and all of the branches and nodes descending from it,
- so it forms a complete game by itself
Examples of Subgames
Each of these green boxes contains a subgame.
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Examples of Subgames
Each of these green boxes contains a subgame.
- The smaller boxes are trivial subgames, because they don’t contain any meaningful choices.
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Examples of Subgames
A game is, technically, a subgame of itself, but it is an improper subgame.
- (Any subgames other than the entire game are proper subgames.)
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Subgame Perfection
So why do we care about subgames?
subgame-perfect Nash equilibrium (aka SPNE), that does not produce the questionable equilibria we saw earlier.
All SPNE are also regular Nash equilibria.
But unique to SPNE, they have to also produce a Nash equilibrium in every subgame
SPNE in the Entry Game
There is only a single proper subgame in the Entry Game.
A SPNE of this game must have a NE in that subgame.
The only NE of the subgame is for the Incumbent to Accommodate.
SPNE in the Entry Game
- Given that the Incumbent Accommodates, the Entrant now faces a choice between staying Out, or Entering and being Accommodated.
- The Entrant’s best response is to Enter, and the SPNE is (Enter, Accommodate)
Rationality and Subgame Perfection
- To justify looking for SPNEs instead of NEs, we assume commonly known sequential rationality.
- Players are aware that other players are sequentially rational.
- Knowing that other players are sequentially rational, each player can predict the others’ rational behavior and chooses an action with the highest payoff at each of their decision nodes.
- Sequentially rational players choose their strategy on the fly, one move at a time, instead of all at once at the beginning of the game.
Finding SPNEs: Backward Induction
The method for finding SPNEs is straightforward—we’ve already been using it without spelling it out formally.
- Find all the decision nodes in the lowest row (assuming the tree goes top down)
- Find the optimal choice at each of these nodes, mark the corresponding branch (or erase/cross out all the other branches).
- If you just solved the game’s initial node, you’re done!
- Otherwise, identify all of the decision nodes in the next row up and return to step 2.
At each step, remember to take into account the optimal moves that you’ve already found lower in the tree.
General Method for Finding Nash Equilibria
Introduction
The method for finding the Nash equilibrium in a game with continuous strategies is general and applies to many types of games.
Key steps:
- Label each player’s strategy (e.g., \(x\), \(y\), etc.).
- Define each player’s payoff as a function of all players’ strategies.
- Maximize each player’s payoff by choosing their own strategy, holding the others constant.
Question: How does this method apply to real-world competition scenarios?
Step 1: Labeling Strategies and Payoffs
Suppose we have players 1, 2, and 3, with strategies labeled \(x\), \(y\), and \(z\), and their payoffs \(X\), \(Y\), and \(Z\).
For each player: \[
X = F(x, y, z), \quad Y = G(x, y, z), \quad Z = H(x, y, z)
\]
Example: Price Competition
- \(x\) and \(y\) could represent the prices set by two firms, and the functions \(F\) and \(G\) represent their profits.
Question: Why is labeling strategies and payoffs essential for structuring a game?
Step 2: Maximizing Payoff Functions
Each player chooses their strategy to maximize their own payoff, considering the strategies of the others as fixed.
To do this, we take the partial derivative of each player’s payoff with respect to their own strategy and set it to zero.
Example:
For firm 1: \[
\frac{\partial F(x, y, z)}{\partial x} = 0
\]
Question: What does this partial derivative represent in terms of strategy adjustment?
Step 3: Solving for Best-Response Functions
Solve each equation to find the best-response functions. These tell each player the best strategy given what the other players are doing.
For firm 1, the best-response function could look like: \[
x = f(y, z)
\]
This gives the optimal \(x\) for any given values of \(y\) and \(z\).
Step 4: Nash Equilibrium
The Nash equilibrium occurs where all players’ best-response functions intersect. Solve the system of equations: \[
x = f(y, z), \quad y = g(x, z), \quad z = h(x, y)
\]
At this point, no player has an incentive to change their strategy.
Example:
In price competition, the Nash equilibrium is where both firms set prices that maximize their respective profits.
Interactive Question: Why is it important that no player has an incentive to deviate at Nash equilibrium?
Step 5: Handling Multiple Solutions
Some games may have multiple Nash equilibria, while others may have none in pure strategies.
Discussion:
- Multiple equilibria can occur in coordination games.
- Games without equilibria in pure strategies might require us to consider mixed strategies (covered in a later chapter).
Recap: General Steps for Nash Equilibrium
- Label each player’s strategies and payoffs.
- Maximize each player’s payoff with respect to their own strategy.
- Solve the system of best-response functions.
- Check for multiple equilibria and the possibility of mixed strategies.
Critical Discussion of the Nash Equilibrium Concept
Discussion of Nash Equilibrium
Now that you’ve learned what it means and seen it applied to several different games, what do you think about the concept of Nash Equilbrium?
- Does it make intuitive sense? Do you think people really reason through their best response to any possible situation?
- Is there anything relevant to strategic interactions you can think of that is left out of our models so far?
The Appeal of Nash Equilibrium
The Nash equilibrium is based on the idea that every player’s strategy should be the best response to the strategies of others.
Benefits of using NE:
Some Potential Critiques of NE predictions
Some Potential Critiques of NE predictions
IESDS, Best-response functions may not be how players actually think
Nash doesn’t explicitly take risk into account
with ordinal payoffs, we might leave out the incentive to play it safe to avoid very bad outcomes
but if we instead use cardinal payoffs, or expected utilities, we can still extend our models to those situations
Some Potential Critiques of NE predictions
IESDS, Best-response functions may not be how players actually think
Nash doesn’t explicitly take risk into account
NE don’t have to be unique
- If we care predictions, NE doesn’t always tell us what will happen, sometimes there can be multiple possible strategy profiles which are equally stable which means any one could happen depending how the game plays out
Criticism 1: Treatment of Risk
We’ve mentioned before how sometimes the strategies played in NE don’t line up with what might intuitively feel like the ‘safe’ strategies.
Criticism 1: Treatment of Risk
Let’s look an an example:
| A |
2,2 |
3,1 |
0,2 |
| B |
1,3 |
2,2 |
3,2 |
| C |
2,0 |
2,3 |
2,2 |
What’s the NE?
Criticism 1: Treatment of Risk
Let’s look an an example:
| A |
2,2 |
3,1 |
0,2 |
| B |
1,3 |
2,2 |
3,2 |
| C |
2,0 |
2,3 |
2,2 |
What’s the NE?
Criticism 1: Treatment of Risk
| A |
2,2 |
3,1 |
0,2 |
| B |
1,3 |
2,2 |
3,2 |
| C |
2,0 |
2,3 |
2,2 |
But what happens if the other player mistakenly plays C?
Then it might make sense to choose the safe option of C
You could get the same payoff of 2, but ensure that you wouldn’t have a chance to get 0
- The idea of trembling-hand perfection takes this into account by allowing probabilistic errors in strategies.
Criticism 1: Treatment of Risk
What about this game?
| Up |
9, 10 |
8, 9.9 |
| Down |
10, 10 |
-1000, 9.9 |
Let’s suppose that there is a small probability \(p\) that Player B will accidentally play Right > What is Player A’s expected utility of playing Up?
\[ EU_A(Up) = 9(1-p) + 8p \]
What is Player A’s expected utility of playing Down?
\[ EU_B(Down) = 10(1-p) - 1000p \]
So Player A should only play Down if:
\[
\begin{align}
10(1-p) - 1000p > & 9(1-p) + 8p \\
p > & \frac{1}{1009}
\end{align}
\]
Criticism 2: Multiple Equilibria
In some games, multiple Nash equilibria exist. This can create problems because: - It’s unclear which equilibrium the players will choose. - Some equilibria may not make sense in real-life situations.
Example:
| T4 |
2, 1 |
0, 0 |
| Metropol |
0, 0 |
1, 2 |
- In coordination games like Battle of the Sexes, there are multiple stable outcomes.
Criticism 2: Multiple Equilibria
Another Example
Pick one player to be Boston, one player as San Francisco
Here are 9 cities which must be evenly divided between the two players:
- Atlanta
- Chicago
- Dallas
- Denver
- Los Angeles
- New York
- Philidelphia
- Seattle
You win if you evenly split this list between you without communicating, and without overlaps or missed cities
Criticism 2: Multiple Equilibria
Arriving at Specific NE
This last example shows the importance of shared background and culture in forming strategic beliefs for cases with multiple NE
So far, our theory have given us sets of NE which are stable given any set of realized beliefs
But it doesn’t help us exaplain how those beliefs might be formed
Evolutionary Game Theory adds tools to our basic toolkit to try to answer some of these limitations
Criticism 2: Multiple Equilibria
In the later chapters we will develop ideas to help narrow down the more general concept of NE
Refinements to NE
A refinement is a restriction on the more general assumptions needed for Nash Equilibria
Examples:
- mixed-strategy Nash equilibrium allows for probabalistic plays
- subgame-perfect Nash equilibria aka rollback equilibria are refinements in sequential games
- perfect Bayesian equilibrium has to deal with informations asymmetries
Criticism 3: Requirements of Rationality
One major criticism is that Nash equilibrium assumes:
Counter-Argument:
- What are some situations you can think of where these assumptions might break down?
Criticism 3: Requirements of Rationality
Consider this game:
| R1 |
0, 7 |
2, 5 |
7, 0 |
| R2 |
5, 2 |
3, 3 |
5, 2 |
| R3 |
7, 0 |
2, 5 |
0, 7 |
What is the NE?
Criticism 3: Requirements of Rationality
Consider this game:
| R1 |
0, 7 |
2, 5 |
7, 0 |
| R2 |
5, 2 |
3, 3 |
5, 2 |
| R3 |
7, 0 |
2, 5 |
0, 7 |
What is the NE?
(R2, C2) is the only NE in this game
For this strategy profile, both players’ beliefs about the other player’s strategy are correct.
Column believes that Row will choose R2, so she chooses her best response of C2
Row believes that Column will choose C2, so he chooses his best response of R2
So in NE, each player’s belief matches their opponent’s best response
Criticism 3: Requirements of Rationality
| R1 |
0, 7 |
2, 5 |
7, 0 |
| R2 |
5, 2 |
3, 3 |
5, 2 |
| R3 |
7, 0 |
2, 5 |
0, 7 |
But we could use similar logic for this game even when beliefs are incorrect
We can justify Row playing R1 if she believes that Column will play C3
Row could beleive this if she believes that Column believes she is playing R3
And if she also thinks that Column believes that she is playing R3 because Column believes that Row believes that Column is playing C1
Rebuttal to Critics: Why Not Nash Equilibrium?
Game theorist Roger Myerson suggests: - “Why not Nash equilibrium?” If someone proposes an alternative strategy profile, that profile must be evaluated. - If it isn’t a Nash equilibrium, one of the players can improve by changing their strategy, disproving the alternative.
Interactive Question: How does this defense of Nash equilibrium answer critics’ concerns?
Alternative Concepts
Criticism has led to the development of alternative solution concepts, such as: - Evolutionary Stable Strategies (ESS): Strategies that persist over time through evolutionary pressure. - Correlated Equilibrium: Players coordinate their strategies based on signals they receive from a common source.
Conclusion
While Nash equilibrium has clear limitations, it remains a central concept in game theory. Alternative solution concepts may provide better insights into certain situations, but Nash equilibrium is still the starting point for analyzing many strategic games.
