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Game Theory and Vehicle Dynamic

Game Theory and Vehicle Dynamic
by

Saeid Khosravani

on 25 March 2013

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Transcript of Game Theory and Vehicle Dynamic

The Nash Equilibrium Notes Multi-Objective & Game-Theory (cc) photo by Metro Centric on Flickr (cc) photo by Franco Folini on Flickr (cc) photo by jimmyharris on Flickr (cc) photo by Metro Centric on Flickr the study of mathematical model of conflict and cooperation between intelligent rational decision-makers ... small Mechanical and Mechatronics Engineering Game Theory
&
Vehicle Control Outline Game Theory
Nash Equilibrium
Differential Game Theory
Optimal Control and Game Theory
Multi Objective Optimization
Vehicle Control Definition John von Neuman (1903-1957) co-authored, Theory of Games and Economic Behavior, with Oskar Morgenstern in 1940s, establishing game theory as a field.

John Nash (1928 - ) developed a key concept of game theory (Nash equilibrium) which initiated many subsequent results and studies
Since 1970s, game-theoretic methods have come to dominate microeconomic theory and other fields

Nobel prizes
Nobel prize in Economic Sciences 1994 awarded to Nash, Harsanyi
(Bayesian games) and Selten (subgame perfect equilibrium)
2005, Auman and Schelling got the Nobel prize for having enhanced
our understanding of cooperation and conflict through game theory
2007 Leonid Hurwicz, Eric Maskin and Roger Myerson won Nobel Prize
for having laid the foundations of mechanism design theory. History Representation of games The games studied by game theory are well-defined mathematical objects.

A game consists of a set of players, a set of moves
(or strategies) available to those players, and a specification of payoffs
for each combination of strategies. Who is interacting? What are their options? What are their incentives? Information: What do they know? Rationality: How do they think? Further Explanation Discipline aiming at modeling situations in which actors have to make decisions which have mutual, possibly conflicting, consequences Example: should a company invest in a new plant, or enter a new market, considering that the competition may make similar moves? Most widespread kind of game: non-cooperative (meaning that the players do not attempt to find an agreement about their possible moves) and cooperative (two or more player attempt to obtain maximum benefit working together - In cooperative game theory, binding agreements are allowed and the unit of analysis is the group or coalition. ) Different Games Non-cooperative
static games Repeated games Stochastic games Cooperative Games Dynamic games ODE for state,
Optimization utility over time,
HJB and dynamic programming Evolutional game Dominant strategy is a player's best strategy, i.e., a strategy that yields the highest utility for the player regardless of what strategies the other players choose. Quiet Fink Quiet Fink 1,1 4,0 0,4 3,3 Player 1 Player 2 Prisoner’s dilemma Strict dominance

strictly best strategy, for any strategy of the other player(s) . Two suspects in a major crime held for interrogation in separate cells

If they both stay quiet, each will be convicted with a minor
offence and will spend 1 year in prison

If one and only one of them finks, he will be freed and used as a witness against the other who will spend 4 years in prison

If both of them fink, each will spend 3 years in prison Components of the Prisoner’s dilemma

Rational Players: the prisoners

Strategies: Stay quiet (Q) or Fink (F)

Solution: What is the Nash equilibrium of the game?

Representation in Strategic Form Prisonner's
Delimma Different
Nash Equilibrium Quiet Fink Quiet Fink 1,1 4,0 0,4 3,3 Player 1 Player 2 Components of the Prisoner’s dilemma

Rational Players: the prisoners

Strategies: Stay quiet (Q) or Fink (F)

Solution: What is the Nash equilibrium of the game?

Representation in Strategic Form Quiet Fink Quiet Fink 2,3 0,0 0,0 3,2 Player 1 Player 2 Prisoner’s dilemma Strict dominance

strictly best strategy, for any strategy of the other player(s) . Components of the Prisoner’s dilemma

Rational Players: the prisoners

Strategies: Stay quiet (Q) or Fink (F)

Solution: What is the Nash equilibrium of the game?

Representation in Strategic Form Does the Nash equilibrium always exist?

Is it efficient ?

One measure of efficiency is Pareto optimality.

In some references a strategy profile that achieves a Pareto optimal payoff distribution can sometimes be referred to as a Pareto optimal strategy Pareto Optimal
Nash Equilibrium Nash Equilibrium (3,3) is not a good equilibrium because there is no guarantee that P1 also fink.
On the other hand even if P2 be sure that P1 will fink, the best action is to Fink. Multiple
Nash
Equilibrium Two-person, zero-sum differential game basic idea: two players control the dynamics of some evolving system, and one tries to maximize, the other to minimize, a payoff functional that depends upon the trajectory. Cooperative Game Theory Players have mutual benefit to cooperate

Bargaining situation
A number of individuals have a common interest to cooperate but a conflicting interest on how to cooperate

A cooperative game means that the players agree to form
coalitions under the expectation that, by working together
a mutually beneficial outcome can be obtained. Preliminary
definitions Multiobjective optimization
and
Game Theory Saeid Khosravani convert multi-objective optimization problem to a single-objective optimization problem (i.e. considering one of the cost function as the leader and the other as its constraints)
transform multi-objective optimization problem into multiple single-objective optimization problems
Minimax theory
Goal Programming
Weighted Sum
Evolutionary Algorithms
Game theory General Form Linear Function Extension to
Non-Linear Open Problem Game Theory
and
Vehicle Control Vehicle Dynamic Prof. Ahmadian’s group used Co-operative differential Game theory to find Nash equilibrium of cost function (quadratic form) , and consequently optimal solution of the problem.
Although their simulation shows improvement in control of the vehicle, one may ask, if we can model vehicle system control and driver as players of a Game?!
What are utilities and payoffs?
The driver can not be a player because when control strategy is on, he can not decide whether use the controller or not?
It is quite hard to consider controller as a player as well!
Does Optimal Control theory provide us the best solution for this circumstance?! Discussion Research Up to now there are no nice (i.e. easily checkable) conditions
ensuring the existence of solution for Riccati Equations.

On the other hand numerical solutions approaches are time consuming
and may not be efficient and applicable in real-time.


Driver models are usually based on preview of the road ahead Discussion Driver / VSC interaction model The Driver’s Steering input and the Controller’s compensated yaw moment are defined as two dynamic players of the game “Vehicle Stability” Modeling as a LQR/LQT Nonlinear VEHICLE MODEL for simulation
Linear model for Control Course Notes on the internet.
T. Basar and G.J. Olsder, Dynamic Non-cooperative Game Theory, 2nd edition, SIAM Classics,. 1999
Tamaddoni, S., Taheri, S., and Ahmadian, M., "Linear Quadratic Game Theory Approach to Optimal Preview Control of Vehicle Lateral Motion," SAE Int. J. of Passeng. Cars – Mech. Syst. 4(1)
M. A. Johnson, 2011, A Differential Game-Based Control methods for uncertain continuous time Nonlinear systems, University of Florida References Thank you s4khosra@uwaterloo.ca Non-cooperative static games Repeated
games Stochastic
games Cooperative
Games Dynamic
games Best Choice for P2 http://www.engineering.uwaterloo.ca/~s4khosra/
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