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The Gittins index is a measure of the reward that can be achieved by a process evolving from its present state onwards with the probability that it will be terminated in the future. It is a real scalar value associated to the state of a stochastic process with a reward function and with a probability of termination.

Terminology

To illustrate the theory we can take two examples from a developing sector, such as from electricity generating technologies: wind power and wave power. If we are presented with the two technologies when they are both proposed as ideas we cannot say which will be better in the long run as we have no data, as yet, to base our judgments on. It would be easy to say that wave power would be too problematic to develop as it seems easier to put up lots of wind turbines than to make the long floating generators, tow them out to sea and lay the cables necessary.

If we were to make a judgment call at that early time in development we could be condemning one technology to being put on the shelf and the other would be developed and put into operation. If we develop both technologies we would be able to make a judgment call on each by comparing the progress of each technology at a set time interval such as every three months. The decisions we make about investment in the next stage would be based on those results.

In a paper in 1979 called Bandit Processes and Dynamic Allocation Indices John C. Gittins suggests a solution for problems such as this. He takes the two basic functions of a "Scheduling Problem" and a "Multi-armed bandit" problem and shows how these problems can be solved using Dynamic allocation indices. He first takes the "Scheduling Problem" and reduces it to a machine which has to perform jobs and has a set time period, every hour or day for example, to finish each job in. The machine is given a reward value, based on finishing or not within the time period, and a probability value of whether it will finish or not for each job is calculated. The problem is "to decide which job to process next at each stage so as to maximize the total expected reward." He then moves on to the "Multi–armed bandit problem" where each pull on a "one armed bandit" lever is allocated a reward function for a successful pull, and a zero reward for an unsuccessful pull. The sequence of successes forms a Bernoulli process and has an unknown probability of success. There are multiple "bandits" and the distribution of successful pulls is calculated and different for each machine. Gittins states that the problem here is "to decide which arm to pull next at each stage so as to maximize the total expected reward from an infinite sequence of pulls."

Gittins says that "Both the problems described above involve a sequence of decisions, each of which is based on more information than its predecessors, and these both problems may be tackled by dynamic allocation indices."

Definition

In applied mathematics, the "Gittins index" is a real scalar value associated to the state of a stochastic process with a reward function and with a probability of termination. It is a measure of the reward that can be achieved by the process evolving from that state on, under the probability that it will be terminated in future. The "index policy" induced by the Gittins index, consisting of choosing at any time the stochastic process with the currently highest Gittins index, is the solution of some stopping problems such as the one of dynamic allocation, where a decision-maker has to maximize the total reward by distributing a limited amount of effort to a number of competing projects, each returning a stochastic reward. If the projects are independent from each other and only one project at a time may evolve, the problem is called multi-armed bandit and the Gittins index policy is optimal. If multiple projects can evolve, the problem is called Restless bandit and the Gittins index policy is a known good heuristic but no optimal solution exists in general. In fact, in general this problem is NP-complete and it is generally accepted that no feasible solution can be found.

History

Questions about the optimal stopping policies in the context of clinical trials have been open from the 1940s and in the 1960s a few authors analyzed simple models leading to optimal index policies, but it was only in the 1970s that Gittins and his collaborators demonstrated in a markovian framework that the optimal solution of the general case is an index policy whose "dynamic allocation index" is computable in principle for every state of each project as a function of the single project's dynamics.

Soon after the seminal paper of Gittins, Peter Whittle demonstrated that the index emerges as a lagrangian multiplier from a dynamic programming formulation of the problem called retirement process and conjectured that the same index would be a good heuristic in a more general setup named Restless bandit. The question of how to actually calculate the index for Markov chains was first addressed by Varaiya and his collaborators with an algorithm that computes the indexes from the largest first down to the smallest and by Chen and Katehakis who showed that standard LP could be used to calculate the index of a state without requiring its calculation for all states with higher index values. LCM Kallenberg provided a parametric LP implementation to compute the indices for all states of a Markov chain. Further, Katehakis and Veinott demonstrated that the index is the expected reward of a Markov decision process constructed over the Markov chain and known as Restart in State and can be calculated exactly by solving that problem with the policy iteration algorithm, or approximately with the value iteration algorithm. This approach also has the advantage of calculating the index for one specific state without having to calculate all the greater indexes and it is valid under more general conditions. A faster algorithm for the calculation of the indexes has been obtained in 2004 by Sonin as a consequence of his elimination algorithm for the optimal stopping of a Markov chain. In this algorithm the termination probability of the process may depend on the current state rather than being a fixed factor. A faster algorithm has been proposed in 2007 by Niño-Mora by exploiting the structure of a parametric simplex to reduce the computational effort of the pivot steps and thereby achieving the same complexity as the gaussian elimination algorithm.

Mathematical definition

Dynamic allocation index

The classical definition by Gittins et al. is:

$\nu (i)={\sup }_{\tau >0}\frac{{⟨\sum _{t=0}^{\tau -1}{\beta }^{t}R[Z(t)]⟩}_{Z(0)=i}}{{⟨\sum _{t=0}^{\tau -1}{\beta }^t⟩}_{Z(0)=i}}$

where $Z(\cdot )$ is a stochastic process, $R(i)$ is the utility (also called reward) associated to the discrete state $i$, $\beta <1$ is the probability that the stochastic process does not terminate, and $⟨\cdot {⟩}_c$ is the conditional expectation operator given c:

$⟨X{⟩}_c\doteq \sum _{x\in \chi }xP\{X=x|c\}$

with $\chi$ being the range of X.

Retirement process formulation

The dynamic programming formulation in terms of retirement process, given by Whittle, is:

$w(i)=\inf \{k:v(i,k)=k\}$

where $v(i,k)$ is the value function

$v(i,k)={\sup }_{\tau >0}{⟨\sum _{t=0}^{\tau -1}{\beta }^{t}R[Z(t)]+{\beta }^{t}k⟩}_{Z(0)=i}$

with the same notation as above. It holds that

$\nu (i)=(1-\beta )w(i).$

Restart-in-state formulation

If $Z(\cdot )$ is a Markov chain with rewards, the interpretation of Katehakis and Veinott (1987) associates to every state the action of restarting from one arbitrary state $i$, thereby constructing a Markov decision process $M_i$.

The Gittins Index of that state $i$ is the highest total reward which can be achieved on $M_i$ if one can always choose to continue or restart from that state $i$.

$h(i)={\sup }_{\pi }{⟨\sum _{t=0}^{\tau -1}{\beta }^{t}R[Z^{\pi }(t)]⟩}_{Z(0)=i}$

where $\pi$ indicates a policy over $M_i$. It holds that

$h(i)=w(i)$.

Generalized index

If the probability of termination $\beta (i)$ depends on the state $i$, a generalization introduced by Sonin (2008) defines the Gittins index $\alpha (i)$ as the maximum discounted total reward per chance of termination.

$\alpha (i)={\sup }_{\tau >0}\frac{R^{\tau }(i)}{Q^{\tau }(i)}$

where

$R^{\tau }(i)={⟨\sum _{t=0}^{\tau -1}R[Z(t)]⟩}_{Z(0)=i}$

$Q^{\tau }(i)={⟨\sum _{t=0}^{\tau -1}\prod _{j=0}^t\beta [Z(j)]⟩}_{Z(0)=i}$

If ${\beta }^t$ is replaced by $\prod _{j=0}^t\beta [Z(j)]$ in the definitions of $\nu (i)$, $w(i)$ and $h(i)$, then it holds that

$\alpha (i)=h(i)=w(i)$

$\alpha (i)\ne k\nu (i),\mathrm{\forall }k$

this observation leads Sonin to conclude that $\alpha (i)$ and not $\nu (i)$ is the "true meaning" of the Gittins index.

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
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  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

External links

• [1] Matlab/Octave implementation of the index computation algorithms

• Cowan, R. Tortoises and Hares: Choice among technologies of unknown merit

• Bertsimas, D. and Niño-Mora, J. (1996). Conservation laws, extended polymatroids and multiarmed bandit problems: a polyhedral approach to indexable systems. Math. Oper. Res. 29 (1), 162–181.