Firming Up the Foundations of Game Theory
Skip to content
Podcast | Insight Unpacked Season 1: Extraordinary Brands and How to Build Them
Strategy Economics May 1, 2011

Firming Up the Foundations of Game Theory

Elucidating the role of information in strategic interactions

Based on the research of

Olivier Gossner

Ehud Kalai

Robert Weber

“Game theory,” says Ehud Kalai, a professor of managerial economics and decision sciences at the Kellogg School of Management, “is a mathematical tool to deal with interactions that involve incentives.” When they apply their skills to sociological, economic, or even biological situations, game theorists hope to gain an understanding of the underlying processes involved in those situations. And although game theory traces its history back to the 1920s, it is continually open to refinements in its meaning and application.

Add Insight
to your inbox.

A significant strand of current research in the field involves the information available to individuals involved in games, an inclusive noun that refers to business deals, negotiations, and other enterprises as well as simple contests of mental skill. Following what has become a tradition in Kellogg’s Managerial Economics & Decision Sciences department, Kalai has made significant contributions to understanding the flow of so-called Bayesian games, in which the participants have what game theorists call “differential information.”

A recent paper coauthored by Kalai, Robert Weber, a professor of managerial economics and decision sciences at the Kellogg School, and Oliver Gossner of the Paris School of Economics and the London School of Economics focuses on two related aspects of differential information: information independence and common knowledge. The project outlined in the paper involves highly technical mathematical operations. According to Kalai, it contributes “a foundational part of game theory.”

Who Knows What?

When two or more individuals undertake any sort of negotiation or competition—for example, selling a house, bidding in an auction, or applying for one of the limited places in a popular college course—proponents of game theory regard them as opponents in a game. The ways in which opponents play a particular game depend critically on the amount of information each possesses—both their own information and the information that they think their opponent has.

To grasp the informational variables in play in game theory, imagine a simple game that matches the seller and potential buyer of a house. Some details about the sale are readily available; they include the asking price, the condition of the house, and the real estate taxes that it incurs. “Game theorists refer to this information as common knowledge variables,” Kalai explains. “Everybody knows the value of the variable, but also everybody knows that everybody knows it, and that everybody knows that everybody knows that everybody knows it, and so on.”

Another situation occurs when players in the game possess unique types of information. This might include how much the seller and the potential buyer like the house in idiosyncratic ways. Here, Kalai says, “the fact that he likes the house a lot does not have any influence on the probability of her liking the house, and vice versa. In such a situation we say that the types are drawn independently of each other, or simply that it is a game with independent types.”

A Condition of Information

In one part of their project, Kalai, Weber, and their European-based colleague studied a condition of information known as independence under common knowledge. This condition means, Kalai explains, that while the information available to the players may not be independent, “there is a common knowledge variable such that beyond this variable the information available to the players is independent of each other.” Put differently, such a variable “disassociates” the information of the players.

To illustrate that situation, Kalai and his colleagues imagine a game in which each player’s mood can affect the way he plays. “Since players care about the behavior of their opponents, it is important for them to know the moods of their opponents,” he says. “But unfortunately, each knows only his own mood.”

“With knowledge of the state of the sun, the players’ moods are independent.” — Kalai

Canny players will undertake a statistical analysis of the likelihood of opponents’ different moods. The analysis can be easy or difficult according to the circumstances.

The condition of independence under common knowledge deals with scenarios that are easy to analyze. Imagine that every player’s mood is equally likely to be happy or depressed if the day is cloudy, and 90 percent likely to be happy and hence just 10 percent likely to be depressed if it is sunny. Overall, the players’ moods are not independent. However, Kalai explains, “with knowledge of the state of the sun, the players’ moods are independent. For example, if a statistician who knows that it is sunny is asked about a player with an unknown mood, he would assign 90 to 10 chance to happy or depressed, regardless of any information about the mood of other players.” Thus, assuming that the state of the sun is common knowledge to all players, all have clear knowledge of the other players’ mood probabilities in every state, whether that be sunny or cloudy.

Now imagine that the mood controller is not the presence or absence of the sun but the level of ozone in the air the players are breathing. This is a more difficult scenario, and one that fails the condition of independence under common knowledge. The moods of the players are now independent, given the current ozone level. But while sunny or cloudy weather is common knowledge, the precise ozone level is not. “Even when a player knows the ozone level, he does not know that his opponents know it, and certainly he does not know what they assume about the knowledge of their opponents,” Kalai explains. “In other words, while the ozone level disassociates the mood probabilities, players have no common knowledge of these probabilities.”

Independence under Common Knowledge and Subjective Independence

Independence under common knowledge is not the only condition that studies of game theory have to consider, though. There is another, seemingly different mathematical condition known as subjective independence. Each player, based on his own information, views the world as though the knowledge possessed by the other players is completely independent of each other. Mathematically, this condition is easy to understand, but the mechanisms behind it were not before the work of Kalai, Weber, and Gossner.

Applying the complex mathematics of game theory to the situation enabled Kalai—himself a Ph.D. mathematician—and his colleagues to gain a deeper understanding of the subjective independence of information. “A main message of our paper is that the hard-to-understand condition of subjective independence may be reinterpreted,” he says. “In games with only two players, this condition is meaningless, since the opponents of a player must be independent of each other; every player has only one opponent. But in games with three or more players, subjective independence is exactly the same as independence under the common knowledge. In other words, in any situation in which every player thinks subjectively that his opponents are independent there must be some objective common-knowledge variable, such as the sun in my example, that disassociates the types of [information possessed by] all the players.” Put simply, Kalai, Weber, and Gossner show that information under common knowledge—a well-understood part of game theory—is equivalent to subjective independence, just from a different mathematical point of view so long as there are three or more players.

And the take-away message of the research: “In the process of discovering and proving the informational relationship,” Kalai continues, “we learned much more about disassociating variables, common knowledge variables, and interesting relationships between them.” To put it more pithily, he adds, “We’re laying a foundation for users of game theory who deal with and analyze strategic interactions.”

Featured Faculty

Professor Emeritus of Managerial Economics & Decision Sciences; Director of the Center for Games and Economic Behavior

Professor Emeritus of Managerial Economics & Decision Sciences

About the Writer
Peter Gwynne is a freelance writer based in Sandwich, Mass.
About the Research

Gossner, Olivier, Ehud Kalai, and Robert Weber. 2009. Information independence and common knowledge.” Econometrica 77: 1317-1328.

Read the original

Most Popular This Week
  1. Your Team Doesn’t Need You to Be the Hero
    Too many leaders instinctively try to fix a crisis themselves. A U.S. Army colonel explains how to curb this tendency in yourself and allow your teams to flourish.
    person with red cape trying to put out fire while firefighters stand by.
  2. What Triggers a Career Hot Streak?
    New research reveals a recipe for success.
    Collage of sculptor's work culminating in Artist of the Year recognition
  3. What’s the Secret to Successful Innovation?
    Hint: it’s not the product itself.
    standing woman speaking with man seated on stool
  4. Which Form of Government Is Best?
    Democracies may not outlast dictatorships, but they adapt better.
    Is democracy the best form of government?
  5. How Much Do Campaign Ads Matter?
    Tone is key, according to new research, which found that a change in TV ad strategy could have altered the results of the 2000 presidential election.
    Political advertisements on television next to polling place
  6. What Went Wrong with FTX—and What’s Next for Crypto?
    One key issue will be introducing regulation without strangling innovation, a fintech expert explains.
    stock trader surrounded by computer monitors
  7. How Are Black–White Biracial People Perceived in Terms of Race?
    Understanding the answer—and why black and white Americans may percieve biracial people differently—is increasingly important in a multiracial society.
    How are biracial people perceived in terms of race
  8. Immigrants to the U.S. Create More Jobs than They Take
    A new study finds that immigrants are far more likely to found companies—both large and small—than native-born Americans.
    Immigrant CEO welcomes new hires
  9. How Experts Make Complex Decisions
    By studying 200 million chess moves, researchers shed light on what gives players an advantage—and what trips them up.
    two people playing chess
  10. Yes, Consumers Care if Your Product Is Ethical
    New research shows that morality matters—but it’s in the eye of the beholder.
    woman chooses organic lettuce in grocery
  11. Why Well-Meaning NGOs Sometimes Do More Harm than Good
    Studies of aid groups in Ghana and Uganda show why it’s so important to coordinate with local governments and institutions.
    To succeed, foreign aid and health programs need buy-in and coordination with local partners.
  12. Product Q&A Forums Hold a Lot of Promise. Here’s How to Make Them Work.
    The key to these online communities, where users can ask and answer questions, is how many questions get useful answers.
    man sits at computer reading Q&A forum
  13. What Went Wrong at AIG?
    Unpacking the insurance giant's collapse during the 2008 financial crisis.
    What went wrong during the AIG financial crisis?
  14. When Do Open Borders Make Economic Sense?
    A new study provides a window into the logic behind various immigration policies.
    How immigration affects the economy depends on taxation and worker skills.
  15. What the New Climate Bill Means for the U.S.—and the World
    The Inflation Reduction Act won’t reverse inflation or halt climate change, but it's still a big deal.
    energy bill with solar panels wind turbines and pipelines
  16. Post-War Reconstruction Is a Good Investment
    Ukraine’s European neighbors will need to make a major financial commitment to help rebuild its economy after the war. Fortunately, as the legacy of the post–World War II Marshall Plan shows, investing in Ukraine's future will also serve Europe's own long-term interests.
    two people look out over a city
  17. How Has Marketing Changed over the Past Half-Century?
    Phil Kotler’s groundbreaking textbook came out 55 years ago. Sixteen editions later, he and coauthor Alexander Chernev discuss how big data, social media, and purpose-driven branding are moving the field forward.
    people in 1967 and 2022 react to advertising
  18. The Political Divide in America Goes Beyond Polarization and Tribalism
    These days, political identity functions a lot like religious identity.
    people engage in conflict with swords
More in Strategy