Utilization factor

Congestion games are a class of games in game theory first proposed by Rosenthal in 1973. In a Congestion game we define players and resources, where the payoff of each player depends on the resources it chooses and the number of players choosing the same resource. Congestion games are a special case of potential games. Rosenthal proved that any congestion game is a potential game and Monderer and Shapley (1996) proved the converse: for any potential game, there is a congestion game with the same potential function.

Motivation

Consider a traffic net where two players originate at point O and need to get to point T. Suppose that node O is connected to node T via connection points A and B, where A is a little closer than B (i.e. A is more likely to be chosen by each player). However, both connection points get easily congested, meaning the more players pass through a point the greater the delay of each player becomes, so having both players go through the same connection point causes extra delay. Good outcome in this game will be for the two players to "coordinate" and pass through different connection points. Can such outcome be achieved? And if so, what will the cost be for each player?

Definition

Discrete congestion games are games with the following components.

• A base set of congestible elements $E$;
• $n$ players;
• A finite set of strategies $S_{i}$ for each player, where each strategy $P\in S_{i}$ is a subset of $E$;
• For each element $e$ and a vector of strategies $(P_{1},P_{2},\ldots ,P_{n})$, a load $x_{e}=\#\{i:e\in P_{i}\}$;
• For each element $e$, a delay function $d_{e}:\mathbb {N} \longrightarrow \mathbb {R}$;
• Given a strategy $P_{i}$, player $i$ experiences delay $\textstyle \sum _{e\in P_{i}}d_{e}(x_{e})$. Assume that each $d_{e}$ is positive and monotone increasing.

Example

Let's consider the following directed graph where each player has two available strategies - going though A or going through B - leading to a total of four possibilities. The following matrix expresses the costs of the players in terms of delays depending on their choices:

The directed graph for a simple congestion game.

Cost Matrix

p1/p2	A	B
A	(5,5)	(2,3)
B	(3,2)	(6,6)

Both (A,B) and (B,A) are pure Nash equilibria in this game.

Existence of Nash equilibria

The existence of Nash equilibria can be shown by constructing a potential function that assigns a value to each outcome. Moreover, this construction will also show that iterated best response finds a Nash equilibrium. Define $\textstyle \Phi =\sum _{e\in E}\sum _{k=1}^{x_{e}}d_{e}(k)$. Note that this function is not the social welfare $\textstyle \sum _{e\in E}x_{e}d_{e}(x_{e})$, but rather a discrete integral of sorts. The critical property of a potential function for a congestion game is that if one player switches strategy, the change in his delay is equal to the change in the potential function.

Consider the case when player $i$ switches from $P_{i}$ to $Q_{i}$. Elements that are in both of the strategies remain unaffected, elements that the player leaves (i.e. $e\in P_{i}-Q_{i}$) decrease the potential by $d_{e}(x_{e})$, and the elements the player joins (i.e. $e\in Q_{i}-P_{i}$) increase the potential by $d_{e}(x_{e}+1)$. This change in potential is precisely the change in delay for player $i$, so $\Phi$ is in fact a potential function.

Now observe that any minimum of $\Phi$ is a pure Nash equilibrium. Fixing all but one player, any improvement in strategy by that player corresponds to decreasing $\Phi$, which cannot happen at a minimum. Now since there are a finite number of configurations and each $d_{e}$ is monotone, there exists an equilibrium.

Continuous congestion games

Continuous congestion games are the limiting case as $n\rightarrow \infty$. In this setup, we consider players as "infinitesimally small." We keep $E$ a finite set of congestible elements. Instead of recognizing $n$ players, as in the discrete case, we have $n$ types of players, where each type $i$ is associated with a number $r_{i}$, representing the rate of traffic for that type. Each type picks a strategy from a strategy set $S_{i}$, which we assume are disjoint. As before, assume that the $d_{e}$ are monotone and positive, but add the assumption that they are continuous as well. Finally, we allow players in a type to distribute fractionally over their strategy set. That is, for $P\in S_{i}$, let $f_{P}$ denote the fraction of players in type $i$ using strategy $P$. Assume that $\textstyle \sum _{P\in S_{i}}f_{P}=r_{i}$.

Existence of equilibria in the continuous case

Note that strategies are now collections of strategy profiles $f_{P}$. For a strategy set $S_{i}$ of size $n$, the collection of all valid profiles is a compact subset of $[0,r_{i}]^{n}$. As before, define the potential function as $\textstyle \Phi =\sum _{e\in E}\int _{0}^{x_{e}}d_{e}(z)\,dz$, replacing the discrete integral with the standard one.

As a function of the strategy, $\Phi$ is continuous: $d_{e}$ is continuous, and $x_{e}$ is a continuous function of the strategy. Then by the extreme value theorem, $\Phi$ attains its global minimum.

The final step is to show that a minimum of $\Phi$ is indeed a Nash equilibrium. Assume for contradiction that there exists a collection of $f_{P}$ that minimize $\Phi$ but are not a Nash equilibrium. Then for some type $i$, there exists some improvement $Q$ over the current choice $P$. That is, $\textstyle \sum _{e\in P}d_{e}(x_{e})>\sum _{e\in Q}d_{e}(x_{e})$. The idea now is to take a small amount $\delta <f_{P}$ of players using strategy $P$ and move them to strategy $Q$. Now for any $x_{e}\in Q$, we have increased its load by $\delta$, so its term in $\Phi$ is now $\textstyle \int _{0}^{x_{e}+\delta }d_{e}(z)dz$. Differentiating the integral, this change is approximately $\delta \cdot d_{e}(x_{e})$, with error $\delta ^{2}$. The equivalent analysis of the change holds when we look at edges in $P$.

Therefore, the change in potential is approximately $\textstyle \delta (\sum _{e\in Q}d_{e}(x_{e})-\sum _{e\in P}d_{e}(x_{e}))$, which is less than zero. This is a contradiction, as then $\Phi$ was not minimized. Therefore, a minimum of $\Phi$ must be a Nash equilibrium.

Quality of solutions and Price of anarchy

Since there exist Nash equilibria in continuous congestion games, the next natural topic is to analyze their quality. We will derive bounds on the ratio between the delay at Nash and the optimal delay, otherwise known as the Price of Anarchy. First, we begin with a technical condition on the delay functions.

Definition The delay is $(\lambda ,\mu )$ smooth if for all $x,y>0$, $yd(x)\leq \lambda yd(y)+\mu xd(x)$.

Now if the delay is $(\lambda ,\mu )$ smooth, $f$ is a Nash equilibrium, and $f^{*}$ is an optimal allocation, then $\textstyle \sum _{e}x_{e}d_{e}(x_{e})\leq {\frac {\lambda }{1-\mu }}\sum _{e}x_{e}^{*}d_{e}(x_{e}^{*})$. In other words, the price of anarchy is $\textstyle {\frac {\lambda }{1-\mu }}$. See these lecture notes for a proof.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Lecture notes of Michal Feldman and Noam Nisan about Potential and congestion games
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

External links

• Lecture notes of Yishay Mansour about Potential and congestion games