1. Game Theory Applications in Network Design
Game theory for WINE
Game theory fr WINE

3. Non-cooperative Games
Bayesian game (Ref-[4])
The game models – static, dynamic, potential, differential and Stackelberg games - were designed based on the governing assumption that all players have the complete information of the game, particularly on the players’ strategies and utility functions.
However, in many real world situations, the complete information of the game may be uncertain or may not be publicly available to other players.
Therefore, it is difficult to know a priori the strategy of players.
Due to this reason, classical non-cooperative game model does not seem to be practical, and cannot be directly applied for a real world.
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4. Non-cooperative Games
Bayesian game
In 1967, John C. Harsanyi developed a highly innovative
analysis of games of incomplete information
and proposed a new game mode.
In this game, a new concept ‘type’ of a player was introduced.
The type is a probability distribution, which is used to express the belief about uncertain or unknown information of the game players.
Type determines the player's payoff function and it is independent and dynamically changed in game stages.
Each player completely knows his own type, but not the types of other players.
Therefore, the incompleteness of information means the uncertainty of the type and utility function of another player.
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5. Non-cooperative Games
Bayesian game
Such a Harsanyi’s developed game is called Bayesian game because of the probabilistic analysis inherent in this game.
Players have initial beliefs about the type of each player where a belief is a probability distribution over the possible types for a player.
According to Bayes’ Rule, players can update their beliefs during the game process.
Therefore, the belief that a player holds about another player's type might change on the basis of the strategies they have played.
The main feature of Bayesian game is to relax the assumption that all information is completely known. This feasible approach can be used to predict the strategic behavior under incomplete information.
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6. Non-cooperative Games
Bayesian game
In a Bayesian game, it is necessary to specify the strategy spaces, type spaces, payoff functions, and beliefs for every player.
A strategy for a player is a complete plan of actions that covers every contingency that might arise for every type that player might be.
A strategy must not only specify the actions of the player given the type that he is, but must also specify the actions that would be taken if he were of another type.
Strategy spaces are defined accordingly. A type space for a player is just the set of all possible types of that player.
The beliefs of a player describe the uncertainty of that player about the types of the other players.
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7. Non-cooperative Games
Bayesian game
Each belief is the probability of the other players having particular types, given the type of the player with that belief, which is defined as Prob(types of other players | type of this player).
A payoff function is a two-place function of strategy profiles and types.
If a player has a payoff function with type t, the payoff received is 𝑢 𝒙 ∗ , 𝑡 , where 𝒙 ∗ is the strategy profile played in the game (i.e. the vector of strategies played).
The formal definition of a Bayesian game is given as follows:
1) Set of players 𝑖∈ 1, 2, , 𝑛 and action set available to player 𝑖 is 𝐴 𝑖 ,
i.e., a i ∈ 𝐴 𝑖 .
2) Sets of possible types for all players 𝑇 𝑖 for 𝑖∈ 1, 2, , 𝑛 , i.e., 𝑡 𝑖 ∈ 𝑇 𝑖 .
3) Let 𝒕= 𝑡 1 , , 𝑡 𝑁 , and 𝒕 −𝑖 = 𝑡 1 , , 𝑡 𝑖−1 , 𝑡 𝑖+1 , , 𝑡 𝑁 .
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8. Non-cooperative Games
Bayesian game
4) 𝒕 is selected according to a joint probability distribution 𝑝 𝒕 on 𝑻= 𝑇 1 ×∙ ∙ ∙× 𝑇 𝑛 .
5) Strategy is defined as 𝑠 𝑖 : 𝑇 𝑖 → 𝐴 𝑖 , and 𝑠 𝑖 𝑡 𝑖 ∈ 𝐴 𝑖 is the action that
type 𝑡 𝑖 of player 𝑖 takes. Payoff is define as 𝑢 𝑖 𝑎 1 , , 𝑎 𝑁 ; 𝑡 1 , , 𝑡 𝑁 .
The Bayesian game proceeds as follows:
i) 𝒕 is selected according to a joint probability distribution 𝑝 𝒕 ,
ii) each player 𝑖 observes realized type 𝑡 𝑖 ,
iii) updates its beliefs based on the Bayesian inference process: each player
calculates with conditional probability of remaining types conditioned on 𝑡 𝑖 = 𝑡 𝑖 ,
iv) denote distribution of 𝒕 −𝑖 conditioned on 𝑡 𝑖 by 𝑝 𝑖 𝒕 −𝑖 𝑡 𝑖 ,
v) finally, players take actions simultaneously.
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9. Non-cooperative Games
Bayesian game
Given strategy 𝑠 𝑖 , type 𝑡 𝑖 of player 𝑖 plays action 𝑠 𝑖 (𝑡 𝑖 ).
With vector of type 𝒕= 𝑡 1 , , 𝑡 𝑛 and strategies 𝑠 1 , , 𝑠 𝑛 , the realized action profile is (𝑠 𝑖 (𝑡 𝑖 ), , 𝑠 𝑛 (𝑡 𝑛 )).
Player 𝑖 of type 𝑡 𝑖 has beliefs about types of other players given by conditional probability distribution 𝑝 𝑖 𝒕 −𝑖 𝑡 𝑖 .
The expected payoff of action 𝑠 𝑖 is
𝒕: 𝑡 𝑖 = 𝑡 𝑖 𝑢 𝑖 ( 𝑠 𝑖 , 𝑠 −𝑖 𝒕 −𝑖 , 𝒕)× 𝑝 𝑖 𝒕 −𝑖 𝑡 𝑖
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10. Non-cooperative Games
Bayesian game
The action 𝑠 𝑖 ( 𝑡 𝑖 ) for player 𝑖 is a best response to 𝑠 −𝑖 𝒕 −𝑖 if and only if for all 𝑠 𝑖 ′ ∈ 𝐴 𝑖
𝒕: 𝑡 𝑖 = 𝑡 𝑖 𝑢 𝑖 ( 𝑠 𝑖 , 𝑠 −𝑖 𝒕 −𝑖 , 𝒕)× 𝑝 𝑖 𝒕 −𝑖 𝑡 𝑖 ≥ 𝒕: 𝑡 𝑖 = 𝑡 𝑖 𝑢 𝑖 𝑠 𝑖 ′ , 𝑠 −𝑖 𝒕 −𝑖 , 𝒕 ×𝑝 𝑖 𝒕 −𝑖 𝑡 𝑖
Each player’s belief, which is the conditional probability distribution 𝒑 𝒊 𝒕 −𝒊 𝒕 𝒊 , is updated periodically based on the Bayesian inference process.
A Bayesian inference process is the process that the game player modifies his prior knowledge about the probability distribution according to the obtained information.
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11. Non-cooperative Games
Bayesian game
The Bayesian inference rule
can be expressed as:
(1) there exists a group of hypotheses 𝐻 1 , 𝐻 2 , , 𝐻 𝑛 relating to event 𝑒
(2) 𝑒 is the evidence, which corresponds to the obtained information that were not
used in computing the prior probability.
(3) 𝑃 𝐻 𝑖 is the prior probability, which is the probability of 𝐻 𝑖 before 𝑒 is observed;
𝑃 𝐻 𝑖 >0, where 𝐻 𝑖 ⋂ 𝐻 𝑗 = ∅, s.t., 𝑖 ≠𝑗.
(4) 𝑃( 𝐻 𝑖 /𝑒) is the posterior probability, which is the probability of 𝐻 𝑖 after 𝑒 is
observed. This is the modified probability of a hypothesis given the observed evidence.
(5) 𝑃(𝑒/ 𝐻 𝑖 ) is the conditional probability, and referred as the happening probability
of 𝑒 when 𝐻 𝑖 happens. It is also known as the likelihood, which indicates the
compatibility of the evidence with the given hypothesis.
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12. Non-cooperative Games
Bayesian game (ref-[2])
Then the Bayesian inference rule formula can be defined as
𝑃 𝐻 𝑖 𝑒)= 𝑃 𝐻 𝑖 ×𝑃 𝑒 𝐻 𝑖 ) 𝑘=1 𝑛 𝑃 𝑒 𝐻 𝑘 )×𝑃( 𝐻 𝑘 )
To enable readers to grasp the fundamental concepts of Bayesian inference, we introduce the buyer-supplier example.
In this example, there are two players – buyer and supplier. Buyer and supplier have their reservation price, 𝑅𝑃 𝑏𝑢𝑦𝑒𝑟 and 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟, respectively.
A player’s reservation price is the player’s threshold of offer acceptability. Typically a reservation price is private to each player, and is different for each player. For example, a supplier’s reservation price is the price such that the supplier player will not accept an offer below this price.
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13. Non-cooperative Games
Bayesian game
A buyer’s reservation price is the price such that the buyer will not accept an offer above this price. Both the buyer and the supplier will make concessions from their initial proposal.
The buyer will increase his initial proposal, while the supplier will decrease his initial proposal.
Eventually, a proposal will be acceptable to both. It is obvious that although the buyer knows his own reservation price, the precise value of 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 is unknown to him.
Nevertheless, the buyer is able to update his belief about 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 based on his interactions with the supplier and on his domain knowledge.
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14. Non-cooperative Games
Bayesian game
As a result of inference process, the buyer is expected to gain more accurate expectation of the supplier's payoff structure and therefore make more advantageous offers.
The buyer’s partial belief about 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 can be represented by a set of hypotheses 𝐻 𝑖 , 𝑖=1, 2, , 𝑛.
For instance, 𝐻 1 can be ‘ 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 = $100’; 𝐻 2 ‘ 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 = $90’. A priori knowledge held by the buyer can be summarized as probabilistic evaluation over the set of hypotheses 𝐻 𝑖 (e.g., 𝑃(𝐻 1 )=0.2, 𝑃(𝐻 2 )= 0.35, . . .).
The Bayesian inference occurs when the buyer receives new signals from the supplier.
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15. Non-cooperative Games
Bayesian game
Along with domain-specific knowledge, these new signals enable the buyer to acquire new insights about 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 in the form of posterior subjective evaluation over 𝐻 𝑖 such as: ‘Usually supplier will offer a price which is above their reservation price by 17%’.
It can be represented by a set of conditional statements of similar form, one of which is shown as follows: 𝑃 𝑒 𝐻 1 )=0.30, where 𝑒 1 represents ‘ Offer 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 = $117’, and 𝐻 1 ‘ 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 = $100’.
Given the encoded domain knowledge in the form of conditional statements and the signal 𝑒 in the form of offers made by the supplier, the buyer can use the standard Bayesian inference formula to revise his belief about 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 .
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16. Non-cooperative Games
Bayesian game
For simplicity, we suppose that the buyer knows that the supplier’s reservation price is either $100 or $90.
In other words, the buyer has only two hypotheses: 𝐻 1 : ‘ 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 = $100’ and 𝐻 2 : ‘ 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 = $90’.
At the beginning of the negotiation, the buyer does not have any other additional information. His a priori knowledge can be summarized as: 𝑃 𝐻 1 =0.5, 𝑃 𝐻 2 =0.5.
In addition, we suppose that the buyer is aware of ‘Suppliers will typically offer a price which is above their reservation price by 17%’, part of which is encoded as: 𝑃 𝑒 𝐻 1 )=0.30 and 𝑃 𝑒 𝐻 2 )=0.05, where 𝑒 1 denotes the event that the supplier asks $117 for the goods under negotiation.
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17. Non-cooperative Games
Bayesian game
Now suppose that the supplier offers $117.
Given this signal and the domain knowledge, the buyer can calculate the posterior estimation of 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 as follows.
𝑃 𝐻 𝑒 1 )= 𝑃 𝐻 1 ×𝑃 𝑒 𝐻 1 ) 𝑃 𝐻 1 ×𝑃 𝑒 𝐻 1 )+ 𝑃 𝐻 2 ×𝑃 𝑒 𝐻 2 ) = 85.7% 𝑃 𝐻 𝑒 1 )= 𝑃 𝐻 2 ×𝑃 𝑒 𝐻 2 ) 𝑃 𝐻 2 ×𝑃 𝑒 𝐻 1 )+ 𝑃 𝐻 2 ×𝑃 𝑒 𝐻 2 ) = 14.3%
Suppose that the buyer adopts a simple negotiation strategy: ‘Propose a price which is equal to the estimated 𝑅𝑃 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 ’.
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18. Non-cooperative Games
Bayesian game
Prior to receiving the supplier’s offer ($117), the buyer would propose $95 (the mean of the 𝑹𝑷 𝒔𝒖𝒑𝒑𝒍𝒊𝒆𝒓 subjective distribution).
After receiving the offer from the supplier and updating his belief about 𝑹𝑷 𝒔𝒖𝒑𝒑𝒍𝒊𝒆𝒓 , the buyer will propose $98.57 instead.
Since the new offer is calculated based on a more accurate estimation of the supplier's utility structure, it might result in a potentially more beneficial final outcome for the buyer and may also help both sides reach the agreement more efficiently.
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19. Non-cooperative Games
Bayesian game
The solution concept of the Bayesian game is the Bayesian Nash equilibrium; a strategy profile (𝑠 𝑖 (𝑡 𝑖 ), , 𝑠 𝑛 (𝑡 𝑛 )) is a Bayesian Nash equilibrium if 𝑠 𝑖 (𝑡 𝑖 ) is a best response to 𝑠 −𝑖 (𝑡 −𝑖 for all 𝑡 𝑖 ∈ 𝑇 𝑖 ) and for all players 𝑖.
In other words, an action specified by the strategy of any given player has to be optimal, given strategies of all other players and beliefs of players.
In a Bayesian game, rational players are seeking to maximize their expected payoff, given their beliefs about the other players.
A Bayesian Nash equilibrium is defined as a strategy profile and beliefs specified for each player about the types of the other players that maximizes the expected payoff for each player given their beliefs about the other players’ types and given the strategies played by the other players.
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20. Non-cooperative Games
Bayesian game
However, the solution concept of Bayesian Nash equilibrium yields an abundance of equilibria in dynamic games, when no further restrictions are placed on players’ beliefs. This makes Bayesian Nash equilibrium an incomplete tool to analyze dynamic games of incomplete information.
To refine the implausible equilibria generated by the Bayesian Nash solution concept, the perfect Bayesian equilibrium solution was developed.
The main idea of perfect Bayesian equilibrium is to refine an abundance of Bayesian Nash equilibria in the same spirit in which subgame perfection equilibrium is to refine implausible Nash equilibria.
The idea of perfect Bayesian equilibrium is profusely used to analyze the game theoretical models that are derived from a wide variety of economic situations.
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21. Non-cooperative Games
Evolutionary game
In a traditional non-cooperative game,
the players are assumed to be rational.
This rationality of the player requires
complete information of game.
However, in reality, this assumption is rarely realistic. From experimental results in economics and the social sciences, people (i.e., game players) occasionally make decisions irrationally.
Even though, the dynamics of the decision-making process can be modeled in extensive form, there is a limitation to capture the fact that a player can observe opponent players’ behaviors learn from this observation, and optimize the strategy selection according to the knowledge gained.
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22. Non-cooperative Games
Evolutionary game
In 1974, Maynard Smith introduced
the fundamental concept of an evolutionary game.
It provides a dynamic framework for analyzing repeated interaction.
At first, evolutionary game has been developed in biological sciences in order to explain the evolution of genetically determined social behavior.
In this game, a population may consist of players genetically ‘programmed’ to play a certain strategy, and who reproduce proportionally to their payoffs.
The payoffs depend on the strategies of the co-players; strategies with high payoff will spread within the entire populations of players. Other strategies which do poorly eventually die off.
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23. Non-cooperative Games
Evolutionary game
Evolutionary games do not require strong rationality.
This approach is suitable for real world situations that involve human beings as players who may not act perfect rational behaviors.
In evolutionary games, the dynamics of interactions among agents in the population can be practically implemented. Therefore, strategy adaptation based on an evolutionary process can be obtained
The changing rate of the players’ selection is defined as Replicator Dynamics (RD).
When a player chooses a strategy, it can change the current game environment and triggers reactions by other players.
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24. Non-cooperative Games
Evolutionary game
After making further changes among players, this interaction mechanism gradually leads the game into a stable state.
The RD describes the evolution in the proportion of each strategy to reach an equilibrium; a specific strategy evolves at a rate equal to the difference between the payoff of that strategy and the average payoff of the whole population .
If the payoff of strategy i is small compared to other strategies, the selection probability for strategy i decreases in proportion to the expected payoff reduction. Therefore, the desirable strategy that will improve player’s payoff is more likely to be selected.
To maximize their expected payoffs, players iteratively change their current strategies and repeatedly interact with other players.
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25. Non-cooperative Games
Evolutionary game
When no individual player can improve his payoff by unilaterally changing his strategy, there is a stable set of strategies. In the jargon of evolutionary game theory, this set is referred to as the evolutionarily stable strategies (ESS).
Under the ESS, the proportions of each strategy do not change in time and can be immune from being changed. It is relevant to the Darwinian evolution mechanism.
To represent the RD for the selection problem, let M be a number of possible strategies and 𝓍 𝑖 is the selection probability for the strategy i.
𝕏 is the M-dimensional vector ( 𝓍 1 … 𝓍 𝑖 … 𝓍 𝑀 ) and 𝓍 𝑖 stands for the variation of 𝓍 𝑖 , which is the RD for strategy i.
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26. Non-cooperative Games
Evolutionary game
J(i, k) is denoted by the expected payoff for a player using strategy i when it encounters a player with strategy k.
J(i, 𝕏) is the payoff for a player using strategy i when it encounters the rest of other players whose strategies are distributed in 𝕏, which can be expressed like as ∑ j (J(i, j) × xj).
Finally, the RD is defined as
𝓍 𝑖 = 𝓍 𝑖 × 𝐽 𝑖, 𝕏 − 𝑗 𝓍 𝑗 ×𝐽 𝑗, 𝕏
= 𝓍 𝑖 × 𝑗 𝓍 𝑗 ×𝐽 𝑖, 𝑗 − 𝑗 𝑘 𝓍 𝑗 ×𝐽 𝑗, 𝑘 × 𝓍 𝑘
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27. Non-cooperative Games
Evolutionary game
The Nash equilibrium is the most common solution concept for non-cooperative games. To obtain this solution, classical game theory usually assumes players are capable of determining the Nash equilibrium that will be played.
However, for some games, this assumption is too stringent because players have incomplete and inaccurate knowledge with bounded rationality.
And, the decision-making process to reach an equilibrium becomes intractable with unreasonable complexity.
Furthermore, multiple Nash equilibrium can co-exist.
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28. Non-cooperative Games
Evolutionary game
In contrast, evolutionary-game theory has been developed to model the behavior of biological agents (e.g., insects and animals). Hence, a strong rationality assumption is not required.
Due to this reason, an evolutionary-game formulation will be suitable for scenarios that involve human beings as agents who may not display hyper-rational behavior. In addition, the solution of the evolutionary game (i.e., ESS) is designed based on an evolutionary process, which is dynamic in nature.
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29. Non-cooperative Games
Evolutionary game
ESS can be obtained with reasonable complexity. Especially, ESS process can serve as a refinement to the Nash equilibrium when multiple Nash equilibria exist. As a new solution concept, ESS is also called evolutionary equilibrium.
Evolutionary game has proven itself to be invaluable in helping to explain many complex and challenging aspects of biology.
Despite its origin and original purpose, the main idea of evolutionary game has emerged as an alternative perspective to classical game theory and become an interesting research field in economists, sociologists, anthropologists, and philosophers.
In particular, evolutionary game model is especially suitable for problems which are non-linear, having large search space (for instance, NP hard problems), multi-dimensional and dynamic problems.
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