Game Theory in Wireless and Communication Networks: Theory, Models, and Applications Lecture 11 Game with Bounded Rationality Zhu Han, Dusit Niyato, Walid Saad, and Tamer Basar

Overview of Lecture Notes Introduction to Game Theory: Lecture 1, book 1 Non-cooperative Games: Lecture 1, Chapter 3, book 1 Bayesian Games: Lecture 2, Chapter 4, book 1 Differential Games: Lecture 3, Chapter 5, book 1 Evolutionary Games: Lecture 4, Chapter 6, book 1 Cooperative Games: Lecture 5, Chapter 7, book 1 Auction Theory: Lecture 6, Chapter 8, book 1 Matching Game: Lecture 7, Chapter 2, book 2 Contract Theory, Lecture 8, Chapter 3, book 2 Learning in Game, Lecture 9, Chapter 6, book 2 Stochastic Game, Lecture 10, Chapter 4, book 2 Game with Bounded Rationality, Lecture 11, Chapter 5, book 2 Equilibrium Programming with Equilibrium Constraint, Lecture 12, Chapter 7, book 2 Zero Determinant Strategy, Lecture 13, Chapter 8, book 2 Mean Field Game, Lecture 14, book 2 Network Economy, Lecture 15, book 2

Overview All players are assumed to be fully rational, which might not true with human and low capable devices. Outline Introduce the concept of bounded rationality and its implication on game theory. develop into the fundamental details of one of the most important frameworks that can capture bounded rationality, prospect theory, which essentially deals with subjective perceptions. conclude by shedding some light on other related notions of bounded rationality. An example

games with bounded rationality In a behavioral game, it is customary to study how certain real-world limits, such as the bounds on the cognitive abilities of a human, can impact the rational tenets of game theory and alter its predicted equilibrium outcomes. For Wireless CPS the natural, cognitive bounds on the rationality of human players (users, administrators, hackers, etc.) that interact with the wireless or cyber-physical system, the limited resources and computational capabilities of certain devices (e.g., IoT sensors) that prevent them from following the conventional, rational path of game theory.

Example: Allais’s Paradox Fear of disappointment. Risk averse.  High Probability 95% chance to win $10,000 100% chance to win $9,499 Hope to avoid loss. Risk seeking 0.95*10000>9499 Low Probability Gain 5% chance to win $10,000 100% chance to win $501 High Probability Lose 95% chance to lose $10,000 100% chance to lose $9,499 0.05*10000<501 Hope of large gain. Risk seeking 0.95*10000>9499 Low Probability 5% chance to lose $10,000 100% chance to lose $501 Fear of large loss. Risk averse 0.05*10000<501

Example: evolutionary game limit on the informational gathering capabilities of the players. due to informational asymmetry and uncertainty, instead of acting as typical optimizers, the players will simply copy strategies from others, based on observable payoffs. In essence, evolutionary games take an “evolutionary biology" approach to optimization, through mutations and natural selection

Prospect Theory Nobel-prize winning PT provides one with mathematical tools to understand how decision making, in real life, can deviate from the tenets of expected utility theory (EUT), a conventional game-theoretic notion which is guided strictly by objective notions of gains and losses, player rationality, conformity to pre-determined decision making rules that are unaffected by real-life perceptions of benefits and risk. people cannot be counted on to always choose optimally among alternatives if merely stating the alternatives differently influences their choices Solution: use prospect-theoretic notions to refine existing game-theoretic mechanisms

Prospect Theory Expected utility theory (EUT) cannot explain the deviations due to end-user subjectivity Prospect theory (PT) [Kahneman,Tversky’79] as a Nobel prize winning theory explains the deviations in monetary transactions: People usually over-weigh low probability outcomes and underweigh their outcomes with high probabilities Loss looms larger than gains Prospect theory has recently been applied in many contexts: Social sciences [Gao’10] [Harrison’09] [Tanaka’16] Communication networks [Li’14] [Yu’14] [Yang’14] [Lee’15] Smart energy management [Wang’14] [Xiao’14]

Fundamentals Tenets of Prospect Theory: Weighting and Framing  

weighting function: Prelec function  

Probability Weighting Functions Probability weighting function models the subjectivity of a player Subjective probability for a player to weigh the outcome with a probability p S-shaped and asymmetrical, ranging in [0,1] Objective weight decreases with the player’s subjective evaluation distortion Prelec function [Prelec’98]: －α=1 －α=0.5

Subjective Perceptions of Utility Functions The Framing Effect utility framing or reference points: each player frames its gains or losses with respect to a possibly different reference point Example: save $10 per month for bill for rich and poor people subjective value functions - concave in gains, convex in losses - over the possible outcomes. Rk is a raw utility reference point and vk is a so-called value function that transforms the raw utility of the player into a PT utility that is framed with respect to a reference point

Framing Effect Illustration of the prospect-theoretic framing effect: how objective utilities are viewed subjectively by human participants. The utility function value changes depending on a certain reference point that highlights the individual perceptions of gains and losses.

Impact of PT on Game Analysis properties may no longer hold under PT requiring new analysis existence of an equilibrium may be jeopardized by PT effects overall operation of optimization mechanisms can be affected traditional equilibrium analysis might not work no general rules for analyzing games under PT exist some application-specific approaches exit

Related Work PT-based channel access game between two subjective end-users in wireless network [Li’12] Wireless operator invests spectrum to users under uncertain spectrum supply using PT [Yu’14] PT-based random access game between two users choosing the transmission probabilities on a radio channel [Li’14] Stackelberg game between the SP offering the bandwidth and subjective end-users to choose services [Yang’14] PT-based microgrids energy trading game in smart grids [Xiao’15]

Other Notions of Bounded Rationality Satisficing: describe a procedure for constructing expectations of how good a potential game solution might reasonably be achieved. Once such a reasonably good solution is achieved, players may stop their search for improvements Satisficing equilibrium: defined as a state at which each player has already reached a “target" utility level or cannot achieve this target by unilaterally changing its action, given the other players actions across all games. a refinement of the idea of a Nash equilibrium, that takes into account the satisficing behavior of the players: instead of seeking a best response, the players will seek a target outcome that satisfies their requirements.

quantal best response players become more likely to make errors as those errors become less costly, a notion that is often known as cost-proportional errors. instead of using a conventional best response, players will react quantally, rather than via strict maximization

quantal response equilibrium (QRE) generalized the Nash equilibrium to the case in which no player can unilaterally improve its utility by using a quantal (rather than a fully rational) best response A QRE is guaranteed to exist for any normal form game and non-negative precision parameter QRE points are not guaranteed to be unique human players will play games differently depending on the magnitudes of the payoffs involved

Summary PT provides a rigorous framework for incorporating bounded rationality in game theory. PT focuses on how game-theoretic decision making is affected by risk and uncertainty, through the use of the weighting and framing effects. The incorporation of PT effects will, however, lead to non-linearities in the utility functions, hence requiring new analysis for the equilibrium or other outcomes of the game. Other types: satisfaction equilibrium and quantal Daniel Kahneman Nobel 2002, "for having integrated insights from psychological research into economic science, especially concerning human judgment and decision-making under uncertainty"