Lecture 8: Putting Yourself in Your Opponent's Cleats

The quiz on this reading has not yet been published.

Last time we introduced strategic environments and looked at dominant strategies: strategies which were the best choice regardless of what the other players in a game were doing.

If the world were made up of dominant strategies, though, game theory wouldn’t be very interesting. This week we’ll look at the fundamental problem of trying to make your best choice based on your beliefs about what the other players are doing. We’re going to build up to the concept of a Nash Equilibrium, in which all players are playing a strategy which is their own best response to the strategies being played by the other player(s).

Before we get to that, though, we need to talk about a couple of important probabilities: beliefs and mixed strategies.

Beliefs and Mixed Strategies

Note: Watson, Chapter 4 is on this subject; so if you want to read a little more about it, I’d recommend checking that out.

There are going to be two important probabilities that come into play in this lecture, both of which can be illustrated by the example of penalty kicks in ~~soccer~~ football.

For those unfamiliar: if a match is tied at the end of play, and it’s a championship (elimination) match so there has to be a winner, it goes to “penalty kicks,” where a sequence of kickers faces a goalkeeper. Here’s a video of the Stanford-Arkansas penalty kicks in the third round of the NCAA championships in 2024:

The players here are the kicker and the goalkeeper. For the sake of argument, let’s assume that there are two strategies for each: the kicker can kick to their left, or to their right; and the keeper can defend the left-hand side of the goal (from the kicker’s perspective), or the right-hand side. (In other words, we’ll use “left” to describe the left-hand side of the goal from the kicker’s perspective; so if they both choose “left,” they’re choosing the same side.)

Let’s further assume that the kicker is better at kicking to her left than her right. Of course, even if the goalkeeper chooses the wrong side, there’s a chance that the ball misses the goal. Let’s assume that probability of scoring is as follows:

If both choose $L$, the kicker scores 60% of the time.
If both choose $R$, the kicker scores 50% of the time.
If the kicker chooses $L$ and the goalkeeper chooses $R$ (dives the other way), the kicker scores 90% of the time.
If the kicker chooses $R$ and the goalkeeper chooses $L$, the kicker scores 80% of the time.

(By the way, I chose these to make the math work out nicely, but if you’re curious, here’s a neat article about the actual probabilities, and how they differ under pressure.)

Finally, we’ll assume that the payoff to the kicker is the probability that the goal is scored, and the payoff to the goalkeeper is the probability the goal is not scored. In that case, we can write the payoff matrix of the game as follows:

layout: OneGameMatrixPlusSidebar: game: player1: name: Kicker strategies: [L,R] player2: name: Goalkeeper strategies: [L,R] payoffs: - [[70,30],[90,10]] - [[80,20],[50,50]] sidebar: controls: - title: Penalty Kicks description: The payoff to the kicker is the probability the goal is scored; the payoff to the goalkeeper is the probability the goal is not scored.

So, which direction should the kicker kick the ball, and where should the goalkeeper defend? To answer that, each one needs to put herself in her opponent’s cleats.

There is clearly not a dominant strategy for either player: if the keeper is going to defend the right-hand side of the goal, the kicker should kick to the left; and vice versa. So, which direction she should kick the ball depends on what she believes about the strategy of the keeper.

Our first probability, therefore, is the probability with which each player believes the other player will choose each of the actions available to him. Let’s write this as a vector $\theta_{-i}$, where $\theta$ indicates the belief and $-i$ indicates it’s about players other than $i$; in this game, therefore, $\theta_{-1}$ would be the belief in the kicker’s mind about where the keeper will defend, while $\theta_{-2}$ would be the belief in the keeper’s mind about where the kicker will kick.

Each belief is a vector of probabilities assigning a likelihood that the other player chooses each strategy in their strategy space. For example, player 2’s strategy space is $S_2 = (L,R)$, so if $\theta_{-1} = (\tfrac{1}{2},\tfrac{1}{2})$, it would mean that the kicker thought it was equally likely that the keeper will defend either side. Likewise, $S_1 = (L,R)$ and if $\theta_{-2} = ({2 \over 3},{1 \over 3})$, it would mean that the keeper thought there was a ${2 \over 3}$ probability that the kicker will go to her left, and a ${1 \over 3}$ probability that she’ll go to her right.

Now, if these players are smart, they won’t always go in the same direction. (If they did, they’d quickly develop a reputation for doing so, and opponents would make easy work of them every time they came up for penalty kicks.) So, rather than choose a “pure” strategy $s_1 = L$ or $s_1 = R$, the kicker might play a mixed strategy: for example, kick to her left ${3 \over 4}$ of the time, and to her right ${1 \over 4}$ of the time. We denote this $\sigma_1 = ({3 \over 4},{1 \over 4})$.

Note that a mixed strategy, like a belief, is a vector of probabilities; but a mixed strategy is a vector of probabilities over a player’s own strategy space, while a belief is a vector of probabilities over strategy space(s) of other player(s). Formally:

Definition (Belief): A belief for player $i$, denoted $\theta_{-i}$, is a probability distribution over the strategy spaces of the other player(s) representing the probability with which player $i$ believes they will play each of their available strategies.

Definition (Mixed strategy): A mixed strategy for player $i$, denoted $\sigma_i$, is a probability distribution over player $i$'s own strategy space representing the probability with which they intend to play each of the (pure) strategies available to them.

Now that we’ve established these definitions, we’re ready to answer two questions:

For a given set of beliefs, what should a player consider doing?
Are there any strategies which a player should not play, regardless of their beliefs?

Best Responses to Beliefs

So: once we’ve formed beliefs about your opponent’s approach to a game, what does that say about what you should do?

Suppose, in the scenario above, the kicker in fact believes that there’s an equal chance the goalkeeper will defend each side of the goal. In that case, the expected utility from choosing each of her two strategies is: $\begin{aligned} \textcolor{3182bd}{u_1}(\textcolor{3182bd}{L}, \textcolor{e6550d}{(\tfrac{1}{2}, \tfrac{1}{2})}) &= \textcolor{3182bd}{70} \times \textcolor{e6550d}{\tfrac{1}{2}} + \textcolor{3182bd}{90} \times \textcolor{e6550d}{\tfrac{1}{2}} = \textcolor{3182bd}{80}\\ \textcolor{3182bd}{u_1}(\textcolor{3182bd}{R}, \textcolor{e6550d}{(\tfrac{1}{2}, \tfrac{1}{2})}) &= \textcolor{3182bd}{80} \times \textcolor{e6550d}{\tfrac{1}{2}} + \textcolor{3182bd}{50} \times \textcolor{e6550d}{\tfrac{1}{2}} = \textcolor{3182bd}{65}\\ \end{aligned}$ Intuitively: given her belief that the keeper will defend to each side half the time:

if she kicks to the left, half the time she’ll get a payoff of 70, and half the time she’ll get a payoff of 90; so her expected payoff is 80.
if she kicks to the right, half the time she’ll get a payoff of 80, and half the time she’ll get a payoff of 50; so her expected payoff is 65. Since $80 > 65$, her best response to those beliefs would be to kick the ball to the left side of the goal.

On the other hand, if she thought there was a 90% chance the keeper was going to defend the left-hand side of the goal, her payoffs would be $\begin{aligned} \textcolor{3182bd}{u_1}(\textcolor{3182bd}{L}, \textcolor{e6550d}{(\tfrac{9}{10}, \tfrac{1}{10})}) &= \textcolor{3182bd}{70} \times \textcolor{e6550d}{\tfrac{9}{10}} + \textcolor{3182bd}{90} \times \textcolor{e6550d}{\tfrac{1}{10}} = \textcolor{3182bd}{72}\\ \textcolor{3182bd}{u_1}(\textcolor{3182bd}{R}, \textcolor{e6550d}{(\tfrac{9}{10}, \tfrac{1}{10})}) &= \textcolor{3182bd}{80} \times \textcolor{e6550d}{\tfrac{9}{10}} + \textcolor{3182bd}{50} \times \textcolor{e6550d}{\tfrac{1}{10}} = \textcolor{3182bd}{77}\\ \end{aligned}$ In this case, since $77 > 72$, her best response to these beliefs would be to kick the ball to the right side of the goal.

With this in mind, we can formally define what a player’s best response to a belief is:

Definition (Best response): Given some belief $\theta_{-i}$ about what the other players are going to play, a strategy $s_i$ is a best response for player $i$ if the expected payoff from playing that strategy is greater than or equal to the expected payoff from playing any other strategy $s_i^\prime$ in their strategy space. That is, $u_i(s_i, \theta_{-i}) \ge u_i(s_i^\prime, \theta_{-i})$ for all $s_i^\prime \in S_i$.

Notice that something interesting happens if she believes the keeper will defend left with exactly an 80%, or ${4 \over 5}$, chance: $\begin{aligned} \textcolor{3182bd}{u_1}(\textcolor{3182bd}{L}, \textcolor{e6550d}{(\tfrac{4}{5}, \tfrac{1}{5})}) &= \textcolor{3182bd}{70} \times \textcolor{e6550d}{\tfrac{4}{5}} + \textcolor{3182bd}{90} \times \textcolor{e6550d}{\tfrac{1}{5}} = \textcolor{3182bd}{74}\\ \textcolor{3182bd}{u_1}(\textcolor{3182bd}{R}, \textcolor{e6550d}{(\tfrac{4}{5}, \tfrac{1}{5})}) &= \textcolor{3182bd}{80} \times \textcolor{e6550d}{\tfrac{4}{5}} + \textcolor{3182bd}{50} \times \textcolor{e6550d}{\tfrac{1}{5}} = \textcolor{3182bd}{74}\\ \end{aligned}$ In this case the expected payoff to either side is exactly the same; so in fact, both $L$ and $R$ are best responses to this set of beliefs. Therefore, when we talk about a player’s possible best responses to a belief, we have to treat this as a set: that is, for any belief $\theta_{-i}$, we can write $BR_i(\theta_{-i})$ as the set of all best responses given that belief. In other words, we can summarize the analysis above as: $\begin{aligned} \textcolor{3182bd}{BR_1}(\textcolor{e6550d}{\tfrac{1}{2}, \tfrac{1}{2} }) &= \{\textcolor{3182bd}{L}\}\\ \textcolor{3182bd}{BR_1}(\textcolor{e6550d}{\tfrac{9}{10}, \tfrac{1}{10} }) &= \{\textcolor{3182bd}{R}\}\\ \textcolor{3182bd}{BR_1}(\textcolor{e6550d}{\tfrac{4}{5}, \tfrac{1}{5} }) &= \{\textcolor{3182bd}{L},\textcolor{3182bd}{R}\}\\ \end{aligned}$ Let’s try to get a little more general, because just plugging in random numbers isn’t a particularly good way to develop a generalizable tool. Let’s say that the kicker believes the goalkeeper will defend to the left with probability $\textcolor{e6550d}{q}$, and to the right with probability $\textcolor{e6550d}{1-q}$; that is, their beliefs can be written by the vector $\theta_2 = \textcolor{e6550d}{(q,1-q)}$. In this case the expected payoff from playing $\textcolor{3182bd}{L}$ or $\textcolor{3182bd}{R}$ is a function of $\textcolor{e6550d}{q}$: $\begin{aligned} \textcolor{3182bd}{u_1}(\textcolor{3182bd}{L}, \textcolor{e6550d}{q}) &= \textcolor{3182bd}{70}\textcolor{e6550d}{q} + \textcolor{3182bd}{90}\textcolor{e6550d}{(1-q)} = \textcolor{3182bd}{90-20q}\\ \textcolor{3182bd}{u_1}(\textcolor{3182bd}{R}, \textcolor{e6550d}{q}) &= \textcolor{3182bd}{80}\textcolor{e6550d}{q} + \textcolor{3182bd}{50}\textcolor{e6550d}{(1-q)} = \textcolor{3182bd}{50+30q}\\ \end{aligned}$ If we plot these out as a function of $q$, we can see that if the kicker believes there’s less than an 80% chance the keeper will defend the left-hand side of the goal, she should kick to the left; and if there’s more than an 80% chance, she should kick to the right:

See interactive graph online here.

We can write the best response, therefore, as a function of the belief $q$: $\textcolor{3182bd}{BR_1}(\textcolor{e6550d}{q}) = \begin{cases} \{\textcolor{3182bd}{L}\} & \text{ if }\textcolor{e6550d}{q < 0.8}\\ \{\textcolor{3182bd}{L},\textcolor{3182bd}{R}\} & \text{ if }\textcolor{e6550d}{q = 0.8}\\ \{\textcolor{3182bd}{R}\} & \text{ if }\textcolor{e6550d}{q > 0.8} \end{cases}$

Best Responses to Strategies

As noted above, a belief is a probability distribution over possible things other players might do. But what if you’re certain they’re going to do something? In other words, what if you place 100% probability on one particular strategy? In that case, we can talk about the best response to a strategy.

For example, consider the Golden Balls example from last class:

layout: OneGameMatrixPlusSidebar: game: player1: name: Contestant 1 strategies: ["\\text{Split}", "\\text{Steal}"] player2: name: Contestant 2 strategies: ["\\text{Split}", "\\text{Steal}"] payoffs: - [[50,50],["\\boxed{0}", "\\boxed{100}"]] - [["\\boxed{100}", "\\boxed{0}"],["\\boxed{0}", "\\boxed{0}"]] sidebar: controls: - title: Golden Balls description: "If the other player is playing Split, you're better off stealing; so Steal is the unique best response to Split. But if they're playing steal, it doesn't matter what you do; so both Split and Steal are best responses to Steal."

Here, the best response if you believe your opponent is playing “Split” is to play “Steal,” because you get 100 rather than 50. Therefore, we might write $BR_i(\theta_{-i} = (1,0)) = \{\text{Steal}\}$ or more simply $BR_i(\text{Split}) = \{\text{Steal}\}$ However, if you believe your opponent is playing “Steal,” you’re going to walk away with nothing no matter what; so your best response set is actually both Split and Steal, since it doesn’t matter what you do. $BR_i(\text{Steal}) = \{\text{Split}, \text{Steal}\}$ What’s interesting about this is that someone actually figured it out, and used it to beat the game. If you think about it, the only way anyone walks away with any money at all is if at least one of the players plays Split; but as we showed last class, this is very much like a prisoners’ dilemma: there’s a weakly dominant strategy to play steal. So how can you get your opponent to play Split? The only way is to completely convince them that you’re going to play Steal…and that’s exactly what Nick Corrigan did. We’ll watch the video in class; it’s epic.

Best responses with continuous strategy spaces

In a model with a continuous strategy space, we’ll often encounter a best response function. That is, if you believe the other player is choosing to play strategy $s_{-i}$, your best response will be a function of that choice: $BR_i(s_{-i})$.

For example, consider the Cournot model. We established last time that the payoff function to firm 2 was $\begin{aligned} \textcolor{e6550d}{\pi_2}(\textcolor{e6550d}{q_2}\ |\ \textcolor{3182bd}{q_1}) &= P(\textcolor{3182bd}{q_1},\textcolor{e6550d}{q_2}) \times \textcolor{e6550d}{q_2} - \textcolor{e6550d}{c_2}(\textcolor{e6550d}{q_2})\\ &= [(14 - \textcolor{3182bd}{q_1}) - \textcolor{e6550d}{q_2}]\textcolor{e6550d}{q_2} - 2\textcolor{e6550d}{q_2}\\ &= \underbrace{14\textcolor{e6550d}{q_2} - \textcolor{3182bd}{q_1}\textcolor{e6550d}{q_2} - \textcolor{e6550d}{q_2^2}}_\text{revenue} - \underbrace{2\textcolor{e6550d}{q_2}}_\text{cost} \end{aligned}$ So, if firm 2 believes (with certainty) that firm 1 will produce some amount $\textcolor{3182bd}{q_1}$, we find firm 2’s best response to this belief by holding $\textcolor{3182bd}{q_1}$ fixed, and taking the derivative with respect to $\textcolor{e6550d}{q_2}$ and setting it equal to zero. In other words, given a fixed value of $\textcolor{3182bd}{q_1}$, firm 2 will maximize its profits where its marginal revenue equals marginal cost, and the solution to that becomes its best response function to $\textcolor{3182bd}{q_1}$: $\begin{aligned} \textcolor{e6550d}{\pi_2^\prime}(\textcolor{e6550d}{q_2}\ |\ \textcolor{3182bd}{q_1}) = \underbrace{14 - \textcolor{3182bd}{q_1} - 2\textcolor{e6550d}{q_2}}_\text{MR} - \underbrace{2}_\text{MC} &= 0\\ 2\textcolor{e6550d}{q_2} &= 12 - \textcolor{3182bd}{q_1}\\ \textcolor{e6550d}{q_2^\star} &= 6 - \tfrac{1}{2}\textcolor{3182bd}{q_1} \equiv \textcolor{e6550d}{BR_2}(\textcolor{3182bd}{q_1}) \end{aligned}$ For example, here’s the graph again showing the profits of the two firms. With $\textcolor{3182bd}{q_1 = 2}$, drag $\textcolor{e6550d}{q_2}$ to see how the profit changes; you should find that it is maximized when $\textcolor{e6550d}{q_2 = 5}$, which is $6 - {1 \over 2} \times 2$. Then change $\textcolor{3182bd}{q_1}$ to some other number, and confirm that firm 2 can’t do any better than to produce $6 - {1 \over 2}\textcolor{3182bd}{q_1}$ (i.e. where its MR curve meets its MC curve):

See interactive graph online here.

Now technically (and this can be important on a homework or test question!) we should specify that the firm will never produce less than zero units of output. Since this expression for $\textcolor{e6550d}{q_2^\star}$ would be negative if $\textcolor{3182bd}{q_1 > 12}$, therefore, we should really write $\textcolor{e6550d}{BR_2}(\textcolor{3182bd}{q_1}) = \begin{cases} 6 - \tfrac{1}{2}\textcolor{3182bd}{q_1} & \text{ if }\textcolor{3182bd}{q_1 \le 12}\\ 0 & \text{ if }\textcolor{3182bd}{q_1 \ge 12} \end{cases}$ Note that we can plot this out: if we let the horizontal axis be $\textcolor{3182bd}{q_1}$ and the vertical axis be $\textcolor{e6550d}{q_2}$, we can plot firm 2’s best response to each possible value of $\textcolor{3182bd}{q_1}$:

See interactive graph online here.

If you really want to peek ahead to Thursday’s lecture, show both best-response curves and think about what it means when they cross…

Best responses to mixed strategies

In section 8.2 we talked about the kicker’s best response to a belief that the goalkeeper would play $L$ with probability $q$ and $R$ with probability $1 - q$.

One related way to think about this is that the goalkeeper actually is playing the mixed strategy $\sigma_2 = (q, 1-q)$, and that the kicker believes with 100% certainty that the goalkeeper is playing that strategy. In this case the same function $\textcolor{3182bd}{BR_1}(\textcolor{e6550d}{q}) = \begin{cases} \{\textcolor{3182bd}{L}\} & \text{ if }\textcolor{e6550d}{q < 0.8}\\ \{\textcolor{3182bd}{L},\textcolor{3182bd}{R}\} & \text{ if }\textcolor{e6550d}{q = 0.8}\\ \{\textcolor{3182bd}{R}\} & \text{ if }\textcolor{e6550d}{q > 0.8} \end{cases}$ describes the best response to the mixed strategy $\sigma_{-i}$ rather than the belief $\theta_{-i}$. But really, it describes the same thing – what the kicker’s best response is to a given probability that the goalkeeper defends the left side of the goal.

Dominated Strategies: Never a Best Response

So, in lecture 7 we talked about what a player should always do – i.e., the situation in which they have a dominant strategy.

We’ve just spent the first half of lecture 8 talking about what a player should sometimes do – i.e., their best response to any belief they might have about what the other players are doing.

Let’s conclude this lecture by talking about what a player should never do – that is, a dominated strategy. We will define a dominated strategy by saying that there is some other strategy (pure or mixed) which dominates it: that is, which yields a higher payoff no matter what the other player does:

Definition (Dominated strategy): A strategy $s_i$ is dominated for player $i$ if there is some other pure strategy $s_i^\prime$ such that $u_i(s_i, s_{-i}) \le u_i(s_i^\prime,s_{-i})$ for all possible opponent strategies $s_{-i}$; or there is some mixed strategy $\sigma_i$ such that $u_i(s_i, s_{-i}) \le u_i(\sigma_i,s_{-i})$ for all possible opponent strategies $s_{-i}$.

As before, we can talk about “weakly dominated” and “strictly dominated” strategies. If your opponent has only two strategies, one way to visualize this is to plot each of your own strategies as a point on a plane, where the horizontal axis measures your payoff if they play one of their two strategies, and the vertical axis measures your payoff if they play the other one.

For example, consider the game below. The graph shows the payoffs to a player from playing $A$ or $B$. The horizontal axis measures the payoff if the other player plays $A$, and the vertical axis measures the payoff if the other player plays $B$. You can drag the points around to change the payoffs in the game.

See interactive graph online here.

Strategies which are dominated by a mixed strategy

Now let’s look at a slightly more complicated case: strategies which are dominated by a mixed strategy.

Suppose we go back to our penalty kicks example. Let’s assume that the kicker has a third option: kick it down the middle of the goal ($M$). Let’s say that regardless of whether the keeper chooses to defend the left or the right side of the goal, there is a 72% probability of scoring. Therefore the game becomes

layout: OneGameMatrixPlusSidebar: game: player1: name: Kicker strategies: [L,M,R] player2: name: Goalkeeper strategies: [L,R] payoffs: - [[70,30],[90,10]] - [[72,28],[72,28]] - [[80,20],[50,50]] sidebar: controls: - title: A Middle Option description: "Suppose we add a third option for the kicker: kicking it down the middle (M). It's not dominated by L or R, but it is ever the best choice?"

It’s clear that $M$ isn’t dominated by either the pure strategy $L$ or $R$ for player 1. But is there some mixed strategy that dominates it?

Take a look at the three outcomes if we plot them in a graph similar to the one above, where the horizontal axis represents the kicker’s payoff if the goalkeeper defends $L$, and the vertical axis represents the payoff if the keeper defends $R$. We can see right away that there is no dominance relationship between the pure strategies $L$, $M$, and $R$; $L$ lies above and to the left of $M$, while $R$ lies below and to the right of $M$. Neither lies in the region above and to the right of $M$, which would give a higher payoff no matter what the keeper does.

However, if we draw a line between these two, we get the expected payoffs for various mixed strategies that the kicker could employ: that is, playing $L$ with probability $p$ and $R$ with probability $1 - p$. Play around with $p$, and find the values for which such a mixed strategy dominates $M$!

See interactive graph online here.

In fact, we can solve for this range of $p$ explicitly. If player 1 is choosing a mixed strategy $\sigma_1 = (p,1-p)$, the expected payoff if the keeper defends to the left is $u_1(\sigma_1,L) = 70p + 80(1-p) = 80 - 10p$ which is greater than the payoff of 72 from kicking down the middle if $\begin{aligned} 80 - 10p &> 72\\ 10p &< 8\\ p &< 0.8\end{aligned}$ Likewise, the expected payoff if the keeper defends to the right is $u_1(\sigma_1,R) = 90p + 50(1-p) = 50 + 40p$ which is greater than the payoff of 72 from kicking down the middle if $\begin{aligned} 50 + 40p &> 72\\ 40p &> 22\\ p &> 0.55\end{aligned}$ So $\sigma_1$ strictly dominates $M$ if $0.55 < p < 0.8$.

Relationship between dominance and best response

The concepts of dominance and best response are closely related, and for games like this – i.e., finite two-player games – there’s actually an important result, which can be useful:

Theorem (Best Response and Domination): In a finite two-player game, if there is no belief that would make $s_i$ a best response for player $i$, then $s_i$ must be dominated by some pure or mixed strategy.

We can see what this means if we look at the graph from before with the expected utility from kicking left and kicking right, and add a horizontal line at 72 representing the payoff from playing $M$:

See interactive graph online here.

The fact that the $u_1(M,q)$ curve is sometimes above the $u_1(L,q)$ curve, and sometimes above the $u_1(R,q)$ curve, means that it’s not dominated by either of those pure strategies.

But, the fact that there is no value of $q$ for which the $u_1(M,q)$ is the highest curve means $M$ is never a best response, and must therefore be dominated by some mixed strategy – as, indeed, we saw it was.

Note: see Watson, pp. 57-60 for a more formal exposition of the relationship between dominance and best response.

Iterated deletion of strictly dominated strategies

It makes sense that no rational player should ever play a strictly dominated strategy, since by definition there is always some other strategy available to them that yields a strictly higher payoff, no matter what the other players are doing. So, we can often simplify games by removing strictly dominated strategies, and analyzing what remains.

For example, consider the game below:

layout: OneGameMatrixPlusSidebar: game: player1: name: Player 1 strategies: [T,M,B] player2: name: Player 2 strategies: [L,C,R] payoffs: - [[1,2],[1,1],[8,0]] - [[4,3],[1,4],[3,2]] - [[3,0],[2,1],[8,0]] sidebar: controls: - title: Iterated Deletion of Strictly Dominated Strategies description: "Which strategies are strictly dominated for each player? If you remove them, do more strategies become dominated, even if they weren't before?"

Here, we can see that $C$ strictly dominates $R$ for player 2, because $1>0$, $4>2$, and $1 > 0$; so we can eliminate $R$ as a strategy. This leaves us with the reduced game:

layout: OneGameMatrixPlusSidebar: game: player1: name: Player 1 strategies: [T,M,B] player2: name: Player 2 strategies: [L,C] payoffs: - [[1,2],[1,1]] - [[4,3],[1,4]] - [[3,0],[2,1]] sidebar: controls: - title: Iterated Deletion of Strictly Dominated Strategies, Step 2 description: "The original game with R removed"

Now that $R$ is removed for player 2, $B$ dominates $T$; so we can remove $T$ from the game:

layout: OneGameMatrixPlusSidebar: game: player1: name: Player 1 strategies: [M,B] player2: name: Player 2 strategies: [L,C] payoffs: - [[4,3],[1,4]] - [[3,0],[2,1]] sidebar: controls: - title: Iterated Deletion of Strictly Dominated Strategies, Step 3 description: "...with R and T removed..."

But the fact that $T$ was a possibility was the only thing keeping player 2 from playing $C$ all the time! With that gone, $C$ dominates $L$ for player 2, so we can remove $L$:

layout: OneGameMatrixPlusSidebar: game: player1: name: Player 1 strategies: [M,B] player2: name: Player 2 strategies: [C] payoffs: - [[1,4]] - [[2,1]] sidebar: controls: - title: Iterated Deletion of Strictly Dominated Strategies, Step 4 description: "...with R, T, and L removed..."

…aaaand with that gone, we can see that player 1 will choose $B$ over $M$, meaning we have reduced the game to a single cell!

layout: OneGameMatrixPlusSidebar: game: player1: name: Player 1 strategies: [B] player2: name: Player 2 strategies: [C] payoffs: - [[2,1]] sidebar: controls: - title: Iterated Deletion of Strictly Dominated Strategies, Step 5 description: "...and then there was one."

Iterated deletion of strictly dominated strategies in the Cournot model

We can use the same process to analyze the Cournot game. We established that each firm had the best response function $q_i = BR_i(q_{-i}) = 6 - \tfrac{1}{2}q_{-i}$ From this, we know that neither firm will produce more than $\bar q = 6$. But if 6 is the maximum value of $q_{-i}$, then each firm must produce at least $\underline q = 6 - {1 \over 2} \times 6 = 3$. And if each firm produces at least 3, we know that no firm will produce more than $\bar q = 6 - {1 \over 2} \times 3 = 4.5$. And if no firm produces more than 4.5, each firm must produce at least $\underline q = 6 - {1 \over 2} \times 4.5 = 3.75$…and so on!

Try using the graphs below to remove the strategies from each strategy space. What do you notice about what’s left once that process concludes?

See interactive graph online here.

How rational is your opponent?

The above analysis is cute, but fundamentally flawed. Why? Because people aren’t fully rational. We’ll see why in class. :)

…but that’s enough reading for today. Let’s summarize what we’ve done, because it’s a lot.

Summary

Beliefs and mixed strategies

There are two important probabilities when analyzing games:

Beliefs are a probability distribution over the other players’ strategy spaces, describing how likely they are to play those strategies
A mixed strategy is a probability distribution over your own strategy space, describing how likely you are to play each (pure) strategy.

Best responses

A strategy $s_i$ is a best response to a belief $\theta_{-i}$ if it yields the highest possible expected payoff, given that belief.

More than one strategy may be a best response; so it’s best to talk about the best response set for any given belief. We generally write the best response to a belief as $BR_i(\theta_{-i})$

If a belief places 100% probability on a single (pure) strategy, we can talk about the best response to a strategy, and write $BR_i(s_{-i})$ (for a pure strategy $s_{-i}$) or $BR_i(\sigma_{-i})$ (for a mixed strategy $\sigma_{-i}$)