Lecture 7: Strategic Environments

For thirteen weeks across Econ 50 and 51, we have looked at what economists call the neoclassical model of markets and prices. With a couple of exceptions we have been dealing with price-taking behavior: both consumers and firms take market prices as given, and decide how much of a good to buy or sell.

This is a great model for deciding how much gasoline people buy. They drive by a gas station, see the posted prices, and decide whether to stop or not. If they do stop, they pay the posted price; they don’t walk into the mini mart and start negotiating with the cashier over the cost of a gallon of gas. Furthermore, while the buying decisions of millions of drivers constitute the demand curve for gasoline, the impact any one driver has on the global market for gasoline is infinitesimally small.

However, this isn’t such a great model for looking at the behavior of oil companies. There are only a few major oil companies in the world (Saudi Aramco, ExxonMobil, and Chevron are the biggest). If one of them dramatically increases production, it lowers the price of oil for all of them. If one invests heavily in renewable energy, the others have to decide how to respond: to compete, or to refocus on fossil fuels?

Fundamental to analyzing the behavior of these oil companies is the realization that they are acting strategically: when they make a decision, they’re not just thinking about their customers; they’re also thinking about their competitors. Saudi Aramco thinks about what ExxonMobil and Chevron are doing, and also thinks about how those companies will respond to what Saudi Aramco is doing. We call such an environment a strategic environment; and the study of strategic environments is called game theory.

Why a “game”? Well, think about what a game is.

• It has players.
• It has rules about what those players can do (what their actions are), and when. It may also specify what each player knows when they make each decision.
• It describes what the outcomes are, as a function of player decisions.
• There may or may not be an element of chance in setting up the game, or determining outcomes. For example, a global pandemic might decrease demand for gasoline, or a war in the Persian Gulf might close the Strait of Hormuz.

We can model the strategic interactions between the oil companies by treating each one as a “player.” We can then write down a “game” that describes the things they might do, and then describe the outcome of each possible combination of what they do. For example, we can think of their “action” as being how much oil to bring to market, and the “outcome” – the price of oil – might depend on how much oil altogether is brought to market by all players. And there might be an element of randomness about how their actions (the total quantity) translates into price – for example, demand for oil might drop due to an unforeseeable event, as it did during the Covid-19 pandemic.

When “playing” such a game, it’s usually not enough to just think about yourself: you have to think about what the other players are doing, and develop a strategy for how you are going to approach the game.

What that strategy might look like depends on the nature of the game being played. Specifically, there are two important dimensions on which games differ:

• Timing: In some games, like rock-paper-scissors, players choose their strategies simultaneously, without observing what the other player(s) are choosing. In other games, like chess, players move sequentially, observing the other players’ actions before taking their own. In still others, players play a simultaneous game repeatedly.
• Information. In some games, all relevant information is known to all players – for example, all moves in chess are taken out in the open. But in many games, like poker (in which players are dealt hands that only they see), one or more players has private information.

The remainder of the class will be organized as follows:

• In weeks 4-6, we’ll deal with games of complete information. Today we’ll introduce the concept of a normal-form game, look at a few examples of strategic environments, and introduce our first equilibrium concept: an equilibrium in dominant strategies. Next week we’ll look at simultaneous (or “static”) games, and introduce the concept of Nash equilibrium. In week 6 we’ll introduce timing and look at sequential (or “dynamic”) games, and refine our concept of Nash equilibrium by looking at which equilibria are what we’ll call subgame perfect.
• In weeks 7-8, we will analyze games of incomplete information. In week 7 we’ll look at static games of incomplete information, in which players have private information but act simultaneously on that information. In week 8 we’ll look at dynamic games of incomplete information, in which players might infer other players’ private information from the way they play the game.
• In weeks 9-10, we’ll turn to the relatively new field of mechanism design, which takes everything we’ve learned and flips it around: if we know how people behave in strategic environments, how can we design those environments to encourage the kind of outcomes we’d like to obtain? The key here will be designing incentive structures that work even when people have private information that isn’t known to the designer.

So, things are going to get pretty complicated…but we’ll introduce new concepts one at a time. As long as you keep current with the course, it should all be manageable.

For now, let’s dive in with the simplest kind of game we’ll be looking at: static games of complete information, and how they might be represented by what’s called the normal form.

Describing a Static Game

In a static game, players move simultaneously. Once their moves are revealed, the outcome is determined. Let’s analyze two such games.

Game 7.1: Did all the extra credit, then got graded on a curve

Many classes have a curved grade policy: your grade isn’t dependent on your absolute performance, but on how you do relative to other students in the class.

Let’s say that, feeling particularly evil, I decide to assign an “optional” extra credit assignment in this class. I’ll match you up with a random student, and your grade will depend on who does the extra credit:

• If both of you do the extra credit, you’ll each get a B.
• If neither of you do the extra credit, you’ll also each get a B.
• If you do the extra credit and your partner doesn’t, you’ll get an A and they’ll get a C.
• If your partner does the extra credit and you don’t, you’ll get a C and they’ll get an A.

It’s a little cumbersome to write out all these possibilities, so we often write these outcomes in a matrix form, with “player 1” shown as the “row player” (i.e. each of their actions corresponds to a row) and “player 2” shown as the “column player” (i.e. each of their actions corresponds to a column). We’ll abbreviate the choices as $E$ for extra credit, and $N$ for no extra credit. For additional clarity, we’ll color player 1 in blue and player 2 in orange. With this, we can display the outcomes, as a function of the choices of each player, in matrix or table form:

Now, note that this table shows the outcomes, but it doesn’t necessarily show how you feel about the outcomes. In particular, one imagines that doing extra credit work comes at a cost – you have better things to do than a lot of extra stuff for econ! So, as Taylor Swift quite properly gripes, doing all the extra credit and then getting graded on a curve can be a lousy feeling.

To address this, we assign a payoff (like a level of utility) to each of the outcomes. Let’s assume that your payoff is just your grade in GPA terms: an A is 4, a B is 3, and a C is 2. Let’s further assume that doing extra credit reduces your payoff by the equivalent of half a grade; so if you do the extra credit and get an A, your utility is $4 - 0.5 = 3.5$. We can then fill in the matrix with the payoffs from each combination of actions:

This is a pretty simple game, so let’s use it as an opportunity to introduce some notation.

First, let’s index the players by some number $i$. So we let $\color{3182bd}i = 1$ be player 1, and $\color{e6550d}i = 2$ be player 2. We’ll mainly use these as subscripts to talk about that player’s strategies or payoffs, as you’ll see in a moment.

Each player’s strategy space, which we denote by a capital $S$, is the set of strategies available to them. In this case, the strategies available to the two players are to do the extra credit ($E$) or not ($N$); so $\color{3182bd}S_1 = \{E,N\}$ is player 1’s strategy space, and $\color{e6550d}S_2 = \{E,N\}$ is player 2’s strategy space.

Each player chooses a strategy from their strategy space; we’ll write the chosen strategy for player $i$ as $s_i \in S_i$. For example, if player 1 does the extra credit and player 2 doesn’t, then \(\textcolor{3182bd}{s_1 = E}, \color{e6550d}s_2 = N\) One shortcut, from thinking about a game from a single player $i$’s perspective, is to write $s_i$ for their strategy and $s_{-i}$ for the strategy of the other player(s). So from the perspective of player 1 in this two-player game, $\textcolor{3182bd}{s_i} = \color{3182bd}{s_1}$ and $\textcolor{3182bd}{s_{-i}} = \color{e6550d}s_2$.

A strategy profile, which we denote by a lower-case $s$ without a subscript, lists out which strategy all players are choosing. So, following on with the example where player 1 does the extra credit and player 2 doesn’t, so $\color{3182bd}s_1 = E$ and $\color{e6550d}s_2 = N$, we would write the strategy profile as \(s = (\textcolor{3182bd}{s_1}, \textcolor{e6550d}{s_2}) = (\textcolor{3182bd}E, \textcolor{e6550d}N)\)

Finally, we have the payoff function for each player, which we’ll denote either $u_i(s)$ for people or $\pi_i(s)$ for firms (think: utility and profit). The payoff function maps every possible strategy profile into a real-number payoff. For example, in the matrix above, we have $\color{3182bd}u_1(E, \textcolor{e6550d}N) = 3.5$ and $\color{e6550d}u_2(\textcolor{3182bd}{E}, N) = 2$.

Having established this notation, we’re ready for the formal definition of a normal-form game:

Definition (Normal-Form Game): The normal-form representation of a game consists of a list of $N$ players $i = 1, 2, ... N$; their strategy spaces $S_1, S_2, ... S_N$; and their payoff functions $u_1(s), u_2(s), ..., u_N(s)$.

Game 7.2: Cournot Duopoly

If the strategy space is discrete, as in the case above, then we can display all the information for a normal-form game in a payoff matrix like the one above. However, if the strategy space is continuous – for example, if each player can choose a real number, or a vector of real numbers, as their strategy – then we generally have to list out the payoff functions.

An example of this is one of the earliest “games” analyzed by economists, from a full century before John Nash formalized the subject. In 1838, Antoine Cournot described a situation in which two firms that produce the same product simultaneously and independently choose how much to produce; the market price is then determined by how much they both produce.

Strategy space

In this game the strategy space for each firm is the quantity they produce; i.e., the set of positive real numbers. If we assume there are two firms, firm 1 and firm 2, then firm 1’s strategy will be its quantity $\color{3182bd}q_1$ and firm 2’s strategy will be its quantity $\color{e6550d}q_2$.

Outcomes

The outcome, as a function of the quantities chosen by each of the firms, is the market price $P$. We’ll assume the market price depends on the total quantity produced by both firms, $Q = \textcolor{3182bd}{q_1} + \textcolor{e6550d}{q_2}$. In particular, we’ll assume that demand for this good is given by \(P(Q) = 14 - Q = 14 - (\textcolor{3182bd}{q_1} + \textcolor{e6550d}{q_2})\) For example, suppose we think of these two firms as ExxonMobil and Chevron. Both of these firms will independently choose how much oil to drill; months later, that oil will get to market, and the supply will be perfectly inelastic at $Q = \textcolor{3182bd}{q_1} + \textcolor{e6550d}{q_2}$. The price will then be determined by the intersection of that vertical supply curve with the demand for gasoline:

Payoffs

Finally, let’s assume that any firm is going to try to maximize its profits, which are total revenues (price times quantity) minus total cost of production: \(\pi_i(q_i) = \underbrace{P(q_i,q_{-i}) \times q_i}_\text{revenue} - \underbrace{c_i(q_i)}_\text{cost}\) Let’s assume for now that each firm can produce this good at a constant marginal cost of 2, with no fixed costs; so $c_i(q_i) = 2q_i$. Combining this with our demand function above, this means that each firm’s payoff, as a function of its own quantity $q_i$ and the other firm’s quantity $q_{-i}$, is \(\pi_i(q_i) = (14 - (q_i + q_{-i}))q_i - 2q_i\)

Note that one way of thinking about this profit is that it’s the profit per unit ($P - c$) times the number of units sold by this firm ($q_i$). We can visualize this as a rectangle with height $P - c$ and width $q_i$. For example,

Note that the downward-sloping lines in the above diagrams are called “residual demand curves.” Let’s look at the one in the right-hand panel, representing firm 2’s perspective. Remember that the market price is $P(q_1,q_2) = 14 - (\textcolor{3182bd}{q_1} + \textcolor{e6550d}{q_2})$. Suppose firm 1 produces $\textcolor{3182bd}{q_1=5}$, as shown in the diagram above. Therefore, if firm 2 doesn’t produce anything (sets $\color{e6550d}q_2 = 0$), the market price is just $14 - (\textcolor{3182bd}{5} + \textcolor{e6550d}{0}) = 9$; so the vertical intercept of firm 2’s residual demand curve when $\textcolor{3182bd}{q_1=5}$ is 9. Now, for each unit firm 2 produces, the price drops by 1; so if $\textcolor{3182bd}{q_1=5}$, the price in the market, as a function of firm 2’s choice of $\textcolor{e6550d}{q_2}$ is \(P(q_2\ |\ \textcolor{3182bd}{q_1=5}) = 14 - (\textcolor{3182bd}{5} + \textcolor{e6550d}{q_2}) = 9 - \textcolor{e6550d}{q_2}\) This is the equation of firm 2’s residual demand curve in the right-hand graph above.

Note that if firm 1 produces more (i.e., increases $q_1$), it shifts the residual demand curve for firm 2 down, and vice versa. (You can see this by adjusting $q_1$ and $q_2$ in the graph.) Intuitively, in a market like this, the more one firm produces, the lower the price other firms get for any given quantity. If ExxonMobil drills more oil, in other words, it doesn’t just reduce the price they get at the pump – it also reduces the price their competitors get.

Dominant Strategies and the Prisoners’ Dilemma

The topic for next week is going to be how to think strategically about what the other players in the game are likely to do, and making your best choice based on that belief. But for some games, you don’t have to think that hard: there’s a best strategy regardless of what the other players do. If such a strategy exists, we call it a dominant strategy. Formally,

Definition (Dominant Strategy): A strategy $s_i \in S_i$ is a dominant strategy if its payoff is higher than all other strategies $s_i^\prime \in S_i$ for all possible strategies played by other players $s_{-i} \in S_{-i}$.

For example, look at the grading game (game 7.1):

If your partner does the extra credit, you should as well: you have a choice between doing the extra credit and getting a B, which gives you a payoff of 2.5, or slacking off and getting a C, which gives you a payoff of 2. But if they don’t do the extra credit, you’re also better off doing it – you’d work hard to get an A, which would give you a payoff of 3.5, as opposed to slacking off like they do and getting a B, which would give you a payoff of 3.

Of course, your partner also has a dominant strategy to do the extra credit; so the equilibrium in dominant strategies is that you both get the extra credit, you both get a B, and you both get a payoff of 2.5.

Note that you could both do better – you could both get the same grade of B if you each agreed to not do the extra credit. In other words, the outcome from the strategy profile $(N,N)$ is a Pareto improvement over the outcome from the strategy profile $(E,E)$. But unlike in the Edgeworth Box, our result here from everyone doing what’s best for themselves is the worst outcome, not the best.

Game 7.3: The Prisoners’ Dilemma

This is, of course, an example of the famous “Prisoners’ Dilemma,” which you’ve probably heard of or learned about in a previous course. The story there, to recap, is that two prisoners are suspected of a crime, but the evidence against them is a little weak. They are interrogated in separate rooms, and given the choice to confess ($C$) to their participation in the crime, or to deny ($D$) that they had anything to do with it. If they both confess, they each go to jail for a long time. The outcomes of this are:

• If they both confess, they both go to jail for 5 years.
• If they both deny responsibility, they’re convicted on lesser charges and both go to jail for 1 year.
• If one confesses and the other denies responsibility, the one who confesses gets immunity to testify against the other; he goes free, and the other one is sentenced to 10 years in prison.

If we assume that the payoffs are just the amount of time spent in jail, we can see that there is a dominant strategy to confess:

There are lots of examples of games with this general form! We’ll be looking at a lot of them in the next few weeks, and talk about a few other examples in class.

Weakly dominant strategies

Let’s close by looking at a slightly different situation: that of weakly dominant strategies. We define a weakly dominant strategy as one whose payoff is always at least as high as any other strategy available to a player, regardless of the strategy chosen by the other player(s). This is distinct from a strictly dominant strategy, whose payoff is always higher than every other strategy. Formally,

Definition (Strictly Dominant Strategy): A strategy $s_i$ is a strictly dominant strategy for player $i$ if its payoff is strictly higher than the payoff from any other strategy player $i$ could choose, for all possible strategies played by other players: that is, $$u_i(s_i,s_{-i}) > u_i(s_i^\prime, s_{-i})$$ for all $s_i^\prime \in S_i$ and $s_{-i} \in S_{-i}$

Definition (Weakly Dominant Strategy): A strategy $s_i$ is a weakly dominant strategy for player $i$ if its payoff is no lower than the payoff from any other strategy player $i$ could choose, for all possible strategies played by other players: that is, $$u_i(s_i,s_{-i}) \ge u_i(s_i^\prime, s_{-i})$$ for all $s_i^\prime \in S_i$ and $s_{-i} \in S_{-i}$

To see weak dominance in action, let’s look at two final games: a notoriously backstabbing British game show, and a car auction.

Game 7.4: Golden Balls

The British game show “Golden Balls” involved contestants earning money through some early rounds, which turned into a grand prize for one final round…and something which was almost a prisoners’ dilemma.

In one episode, the contestants amassed about £100,000 in prize money – no small sum! They then had to play the following game: each player secretly chose a ball marked “split” or one marked “steal.” They keep the ball closed and then simultaneously reveal their choice.

• If both players choose “split,” they split the grand prize, earning £50,000 each.
• If one chooses “split” and the other chooses “steal,” the one who chose “steal” gets the entire £100,000, while the one who chose “split” leaves with nothing.
• If they both choose “steal,” they both, as the host says “leave with what you came in with: nothing.”

If we assume that each of them only cares about money, so that their payoff is just the amount they walk away with, then we can write the payoffs of the game as follows:

In this case, playing “steal” is weakly dominant because it’s never worse than playing “split,” and sometimes better. In particular, if the other player is playing “split,” then playing “steal” gets you an additional £50,000; but if they’re playing “steal,” you’re leaving with nothing anyway, so it doesn’t matter what you play.

We’ll see next time how this really can make a difference in how the game is played!

Game 7.5: Pimp My Ride

Suppose you’ve just graduated from Stanford and are headed for a fancy new job in New York City. You won’t be able to have a car in New York, so you need to sell your prized Mazda Miata. How can you ensure you’re getting the highest possible price?

One method is to hold an auction, in which potential buyers submit bids and you choose from among those bids. But how can you be sure you’re getting the highest possible bids? Well, in 1961, William Vickrey showed (among other things) that if you hold a particular kind of auction, each of the bidders will find that it’s a weakly dominant strategy to bid the absolute most they would be willing to pay for your car. Let’s see how that works!

Following the game theoretic framing above, let’s look at strategies, outcomes, and payoffs for auctions in general, and for a particular kind of auction now called a Vickrey auction.

Strategies

In an auction, the strategies open to players depend on how the auction is set up. In the kind of auction that’s usually shown on television, an auctioneer calls out numbers (“do I hear €10,000?”) and participants bid by raising a paddle. In a “silent auction,” there’s usually a piece of paper in which people write larger and larger numbers. Both of those are “open-bid” auctions, in which bids are public information; and a strategy in those kinds of situations would be a pretty complicated set of behavior based on the way you see other people bidding.

There’s another kind of auction, though, called a “sealed-bid” auction, in which bidders secretly write their bid on a piece of paper or submit it online, and no participant sees any other’s bid. In this kind of auction, your strategy is just the amount you bid. A Vickrey auction is this kind of auction; so each bidder $i$’s strategy will just be how much to bid, $b_i$.

Outcomes

The rules of the auction also have to specify who gets the Miata, and how much each person pays.

You might think that the highest bid always wins the auction and gets the Miata, but that isn’t always the case. Many auctions have a reserve price, and if no bid is higher than the reserve, the object goes back to the seller. At a more technical level, there must also be a tie-breaking rule specifying what happens if two people submit the same bid. (For example, we could flip a coin and assign it randomly to one of them.)

Lastly, there are surprisingly many options for how much each person pays! In a first-price auction, the winner pays the amount of their own bid. In an all-pay auction, every bidder pays the amount of their bid. (This may sound absurd, but think about a situation in which multiple companies donate to a politician’s election campaign in hopes of getting a government contract; the one who donates the most gets the contract, but none of the others get their money back! So that’s kind of like an all-pay auction.)

The Vickrey auction we’ll look at is a third type, called a second-price auction. In this auction, the winner pays the amount of the second-highest bid. So if Adam bids €80, Bianca bids €100, and Carlos bids €400, Carlos wins the auction and pays €100.

Payoffs

We will assume that each bidder values the Miata at some amount $v_i$. If they win, and have to pay some amount $b$, their payoff is $u(v_i,b) = v_i - b$. If they lose, their payoff is zero. Note that we don’t assume any extra utility from winning or any utility loss from losing.

In summary, the game we will consider is:

• Strategies: each player $i$ submits bid $b_i$
• The winner is the bidder with the highest bid; they pay the amount bid by the second-highest bidder
• The payoff to the winner is their valuation $v_i$ minus the amount they pay; the payoff to all other bidders is 0.

In this game, it is a weakly dominant strategy for each player to bid their true valuation $v_i$. Why?

Suppose your friend Mirabai really loves your Miata; she values it at €20K. (Let’s just write this as 20, and know that we’re talking about thousands of dollars.) Let’s say she’s choosing between bidding her true valuation, or a little above or below: specifically, 18 or 22. Let’s think about her payoffs if the highest bid other than hers is 17, 19, 21, or 23.

• If the highest bid other than hers is 17, then regardless of whether she bids 18, 20, or 22, she wins the auction and pays 17, so her payoff is $20 - 17 = 3$. So, bidding her true valuation yields the same payoff as bidding 18 or 22.
• If the highest bid other than hers is 19, then if she bids 18 she loses the auction, whereas if she bids 20 or 22 she wins and gets a payoff of $20 - 19 = 1$. So, bidding her true valuation yields a higher payoff than 18, and the same as 22.
• If the highest bid other than hers is 21, then if she bids 18 or 20 she loses the auction, whereas if she bids 22 she wins…but wishes she hadn’t, since her payoff is $20 - 21 = -1$. So, bidding her true valuation yields a higher payoff than 22, and the same as 18.
• Finally, if the highest bid other than hers is 23, she loses regardless, so bidding her true valuation is no worse than bidding 18 or 22. (And, by the way, it’s better than bidding 24!)

While we only picked a few integers, you can see what’s going on here: depending on what the other players’ strategies are, Mirabai’s payoff from bidding 20 is never lower than bidding any other amount, and sometimes higher than every other amount. Hence, it is a weakly dominant strategy!

If you’re interested in this kind of stuff, you should definitely take Paul Milgrom’s Econ 136 class, which covers auctions and lots of other mechanisms. You might also be interested in reading Milgrom’s speech at the 1996 Nobel⊕ Technically, the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel ceremony when Vickrey won the Nobel for this work. He also has an excellent and very readable overview of auction theory. And of course, he won his own Nobel prize in 2020 for his contributions to this field!