Written Report

Researching and Further Studying of Game Theory
Yuxi Weng, Xueying Liu, Lei Wang
July, 22th, 2015
Game theory can be defined as the study of mathematical models of conflict and cooperation
between intelligent rational decision-makers. Game theory provides general mathematical
techniques for analyzing situations in which two or more individuals make decisions that will
influence one another’s profit. As such, game theory offers insights of fundamental importance
for scholars in all branches of the social sciences, as well as for practical decision-makers.
Game theory, Prisoner’s dilemma, John Nash, Nash Equilibrium
1. Introductiong
Game theory is an equal competition between 2 people, who want to make use of their
opponent’s strategy to change them, so that they could achieve success. The thought of Game
theory actually came from a long time ago, Master Sun's Art of War is a famous work at the
history of China, and it is not only about war but also the earliest Game theory book. At the
beginning, game theory focuses on the success or failure in chess, bridge and gamble, thus we
can see that people’s controlling on gaming situation is just based on experiences, not for
Game theory would think about the predictable action or actual one in games, and find
the way to get better.
The research of Game theory nowadays is beginning with Zemelo, Borel and Von
In 1928, Von Neumann proved that the basic principle of game theory, which declared
the birth of game theory. In 1944, Von Neumann and Morgenstern write the An epoch-making
masterpiece together with is called “Games and Economic Behavior”, and they developed the
2-people Game theory to Multiple Game theory, and apply Game theory to field of Economy,
then they set the foundation and theory structure for this subject.
From 1950 to 1951, John Forbes Nash Jr made use of fixed point theorem to prove the
existence of Equantequation, so he made a strong foundation for game theory’s vague
generalization. Nash’s creative paper: “N game theory” (1950), “Non-cooperative Game”
(1951) and so on, gave Nash concept of equilibrium and equilibrium existence theorem.
Otherwise, Reinhard, hazel, John Harsanyi’s research did promotion for the development of
Game theory. Now Game theory has developed nearly completed subject.
Game theory mainly studies the formulation of interaction between the incentive
structures. It is the way to study; something has mathematics problems with competitive
phenomenon. Game theory considers the game's predictable behavior and actual behavior, and
to study their optimization strategy. Biologists use game theory to understand and to predict
some results of evolution.
Game theory has become one of the standard analytic tools of economics. In biology,
economics, international relations, computer science, politics, military strategy and many
other disciplines are widely used.
Included in the basic concept, all of players, action, information, strategy and yields
balanced results. One player, policies, and income are the basic elements. Players, actions and
results are referred to as game rules.
2. Game theory approaches
2.1 Prisoner’s dilemma
Suppose that the police have arrested two people whom they know have committed a
robbery together. Unfortunately, they lack enough evidence to get a jury to convict. However,
they have enough evidence to send each prisoner away for two years of the car theft. The
chief inspector now makes the following offer to each prisoner: If you confess to the robbery,
and he doesn’t confess, then you'll go free and he'll get ten years in prison. If you both confess,
you'll each get 5 years in prison. If neither of you confess, then you'll each get 2 years for the
car theft.
We can consider it as a game and represent the problem faced by both of players on a
single matrix that has their separate choices interact; this is the form of their game:
Table 1: Strategies which can be used for prisoner’s dilemma
Each cell of the matrix gives the payoffs to both players for each combination of actions.
A's payoff appears as the first number of each pair, B's as the second.
So, if both players confess then they each get 5 years in prison. If neither of them
confesses, they each get a payoff 2 years in prison. If A confesses and B doesn't then A goes
free and B gets a payoff 10 years in prison. The reverse situation, in which B confesses and A
refuses, appears in the lower-left cell.
Each one evaluates his or her two possible actions here by comparing their personal
payoff in each column, because this shows which of their actions is preferable. So, let’s
observe: If B confesses then A gets 5 years in prison by confessing and 10 years in prison by
refusing. If B refuses, then A goes free by confessing and gets 2 years in prison by refusing.
Therefore, A is better off confessing regardless of what B does. B, meanwhile, evaluates his
actions by comparing his payoff in each row, and he comes to exactly the same conclusion
that A does.
Wherever one action for a player is superior to his other actions for each possible action
by the opponent, we say that the first action strictly dominates the second one. In the
Prisoner’s Dilemma, confessing strictly dominates refusing for both players. Thus the best
situation is both players will confess, and both will go to prison for 5 years.
2.2 Chicken game
The other interesting game which can be used in construction management is chicken
game. In this game, imagine two young drivers are driving by two fast cars toward each other
in a narrow road. The probability of death for both young drivers is high, if none of them turn
his direction. The priorities of these two drivers in this game are that they will not play the
timid role. So the best payoff is to have your opponent be the chicken. The worst possible
payoff is to crash to each other. So in the matrix for this game, this situation has the least
value. We assign it 1. As mentioned before, the best payoff for each driver is to have his
opponent be the chicken, so we assign it a value 4. The next worst possibility is to be the
chicken, so we assign this a value 2. The last possibility is that both drivers swerve at the
same time. We assign this a value 3. In this strategy they can maintain their pride and life, so
this is preferable to being the chicken. But in these circumstances none of the players neither
will be a loser nor will be a winner.
Player 1
Player 2
Do not swerve
Do not swerve
Table 2: Strategies which can be used for chicken game
According to the matrix above, the game has two Nash equilibrium points that are
(swerve, do not swerve) and (do not swerve, swerve). In addition being the Nash equilibrium,
these two options can be also Pareto optimal points. There is also another optimal Pareto point
that is (swerve, swerve). At this point, both players reach to equality state and the play will
have no loser or winner.
3. Further reading and further study
3.1 John Nash
Nash was born on June 13, 1928, in Bluefield, W.Va., U.S, who is a mathematician. He
earned a doctorate from Princeton University at 22. He began teaching at Massachusetts
Institute of Technology in 1951 but left in the late 1950s because of mental illness; thereafter
he was informally associated with Princeton. Beginning in the 1950s with his influential
thesis “Non-cooperative Games,” Nash established the mathematical principles of game
theory. His theory, known as the Nash solution or Nash equilibrium, attempted to explain the
dynamics of threat and action among competitors. Despite its practical limitations, it was
widely applied by business strategists. He shared the 1994 Nobel Prize in Economics with
John C. Harsanyi and Reinhard Selten. A film version of his life, A Beautiful Mind, won an
Academy Award for best picture.
3.2 Types of theory game
Cooperative / Non-cooperative
A game is cooperative if the players are able to form binding commitments. For instance,
the legal system requires them to adhere to their promises. In non-cooperative games, this is
not possible.
Often it is assumed that communication among players is allowed in cooperative games,
but not in non-cooperative ones. However, this classification on two binary criteria has been
questioned, and sometimes rejected.
Of the two types of games, non-cooperative games are able to model situations to the
finest details, producing accurate results. Cooperative games focus on the game at large.
Considerable efforts have been made to link the two approaches. The so-called Nash-program,
which is the research agenda for investigating on the one hand axiomatic bargaining solutions
and on the other hand the equilibrium outcomes of strategic bargaining procedures, has
already established many of the cooperative solutions as non-cooperative equilibrium.
Zero-sum / Non-zero-sum
Zero-sum games are a special case of constant-sum games, in which choices by players
can neither increase nor decrease the available resources. In zero-sum games the total benefit
to all players in the game, for every combination of strategies, always adds to zero more
informally, and a player benefits only at the equal expense of others. Poker exemplifies a
zero-sum game ignoring the possibility of the house's cut, because one wins exactly the
amount one's opponents lose. Other zero-sum games include matching pennies and most
classical board games including go and chess.
Many games studied by game theorists, including the infamous prisoners' dilemma, are
non-zero-sum games, because the outcome has net results greater or less than zero. Informally,
in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by
Constant-sum games correspond to activities like theft and gambling, but not to the
fundamental economic situation in which there are potential game from trade. It is possible to
transform any game into a possibly asymmetric zero-sum game by adding a dummy player is
often called "the board" whose losses compensate the players' net winnings.
3.3 Some applied uses
Computer science and logic
Game theory has come to play an increasingly important role in logic and in computer
science. Several logical theories have a basis in game semantics. In addition, computer
scientists have used games to model interactive computations. Also, game theory provides a
theoretical basis to the field of multi-agent systems.
Separately, game theory has played a role in online algorithms. In particular, the k-server
problem, which has in the past been referred to as games with moving costs and
request-answer games.
Yao's principle is a game-theoretic technique for proving lower
bounds on the computational complexity of randomized algorithms, especially online
The emergence of the internet has motivated the development of algorithms for finding
equilibrium in games, markets, computational auctions, peer-to-peer systems, and security
and information markets. Algorithmic game theory and within it algorithmic mechanism
design combine computational algorithm design and analysis of complex systems with
economic theory.
Economics and business
Game theory is a major method used in mathematical economics and business for
modeling competing behaviors of interacting agents.
Applications include a wide array of
economic phenomena and approaches, such as auctions, bargaining, mergers & acquisitions
pricing, fair division, duopolies, oligopolies, social network formation, agent-based
computational economics, general equilibrium, mechanism design, and voting systems; and
across such broad areas as experimental economics, behavioral economics, information
economics, industrial organization, and political economy.
This research usually focuses on particular sets of strategies known as "solution
concepts" or "equilibrium". A common assumption is that players act rationally. In
non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies
is Nash equilibrium if each represents a best response to the other strategies. If all the players
are playing the strategies in Nash equilibrium, they have no unilateral incentive to deviate,
since their strategy is the best they can do given what others are doing.
The payoffs of the game are generally taken to represent the utility of individual players.
A prototypical paper on game theory in economics begins by presenting a game that is an
abstraction of a particular economic situation. One or more solution concepts are chosen, and
the author demonstrates which strategy sets in the presented game is equilibrium of the
appropriate type. Naturally one might wonder to what use this information should be put.
Economists and business professors suggest two primary uses (noted above): descriptive and
Application of Game Theory Approach in Solving the Construction Project Conflicts
Azin Shakiba Barougha, Mojtaba Valinejad Shoubia,*, Moohammad Javad Emami Skardib