Difference between revisions of "Geometric Progression"

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''Relations where each change in effort results in the effect being multiplied by a fixed value.''
''Relations where the change in effect due to a change in effort is measured through a fixed multiplier.''
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Geometric Rewards for Investments are those rewards in games where the growth of the reward in not linear but instead grows with every resources used to gain them. Geometric Rewards for Investments discourages players to split resources into several smaller investments in favor of one large investment as the combined rewards of two lesser investments are always smaller than the reward of the larger investment using the same amount of resources.
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[[Geometric Progression]] occurs in games when the growth of effects are not linear but grows by depending on the current strengths of the effects and constant factor, called the ''common ratio''. For factors larger than 1, [[Geometric Progression]] makes each new effort have a greater effect than the previous one and can thus discourages players to split resources into several smaller investments in favor of one large investment as the combined rewards of two lesser investments are always smaller than the reward of the larger investment using the same amount of resources. Factors less than 1 have the opposite effect and encourage players to spread resources.
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For the purpose of this pattern, the common ratio is assumed to be larger than 0 (progression shifting from positive to negative values are rare in games) and differ from 1 (this would simply make extra effort have no effect). See wikipedia<ref name="wiki"/> for more information on geometric progression.
  
 
=== Examples ===
 
=== Examples ===
Example: in Tetris the reward in points for removing more than one row at the time increase in geometric fashion.
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The point reward in [[Tetris]] for removing more than one row at once increases in a geometric fashion. The pot in the [[Poker]] variant [[Guts]] does not have to grow between rounds, but when it grows it grows by a factor 2 or more leading to a [[Geometric Progression]] in its size until it resets.
  
Example: in Bohnanza, a card game involving trade and luck, collecting more similar cards before cashing them in for victory points gives Geometric Rewards for Investments for some types of cards: 2 points for 2 similar cards, 4 points for 3 similar cards and so on.
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The rent one has to pay for landing on another players' property in [[Monopoly]] grows geometrically for the first three houses but tampers of for the fourth and for hotels. Exchanging collected cards for victory points in [[Bohnanza]] follows a [[Geometric Progression]] but in reducing the number of cards needs to get another point rather than providing more points. For example, 4 ''wax beans'' are required to get 1 points, 7 to get 2 points, 9 for 3 points, and 11 for 4 points.
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The first levels for buying skills in the roleplaying system [[GURPS]] has a [[Geometric Progression]] that encourages players to diversify their skill sets.
  
 
== Using the pattern ==
 
== Using the pattern ==
[[Drawing Stacks]] can be used to create [[Geometric Progression]] since the increase for drawing a specific [[Cards|Card]] increases for each card drawn, and this can be used to create implicit [[Time Limits]] ([[Pandemic]] and [[Thunderstone]] uses this together with [[Stack Seeding]] to guarantee that certain cards appear semi-regularly).  
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[[Geometric Progression]] can be included in games through setting progression schemes. These schemes are often applied to [[Resources]] through [[Investments]] and thereby potential to [[Rewards]] and [[Penalties]], but can also directly modify [[Rewards|Reward]] and [[Penalties|Penalty]] schemes as well as how [[Extended Actions]] or [[Vulnerabilities]] function. As something affected by [[Rewards|Reward]] and [[Penalties|Penalty]], [[Scores|Score]] systems can also be modified to make [[Geometric Progression]] present in the calculations when increasing or decreasing scores. [[Sets]] are another potential target for [[Geometric Progression]], here the schemes apply to the number of game elements collected or equipped. An alternative is to consider changing the effectiveness of individual actions which has the same overall effect if the actions are aimed at the same target but requires prioritization of target order if many targets are chosen.
  
[[Extended Actions]]
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Some other patterns have progression quite close enough to geometric progression to be worthy mention. [[Drawing Stacks]] have the characteristic that the chance of drawing a specific [[Cards|Card]] increase for each extra [[Cards|Card]] drawn, and the increase can for all but the first extra [[Cards|Card]] be describes as a multiplier between 1 and 1.5. This can be used to create implicit [[Time Limits]] - [[Pandemic]] and [[Thunderstone]] uses this together with [[Stack Seeding]] to guarantee that certain cards appear semi-regularly.
[[Randomness]]
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[[Drawing Stacks]]
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[[Combos]]
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== Consequences ==
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[[Geometric Progression]] typically provides [[Positive Feedback Loops]] or [[Negative Feedback Loops]] depending on if the common ratio is above 1 or below 1. With [[Positive Feedback Loops]] the pattern creates [[Combos]] and [[Higher-Level Closures as Gameplay Progresses]], and thereby [[Value of Effort]]. [[Value of Effort]] can also be achieved in a similar way with [[Extended Actions]], that of players creating an increasing value of effort as long as the action is maintained since the action becomes more and more powerful or rewarding the longer it is continued. In contrast, with [[Negative Feedback Loops]] players are instead encouraged to seek [[Varied Gameplay]] in order to avoid [[Diminishing Returns]]. Regardless of what type feedback loops created, this changes how players need to consider [[Collecting]] and [[Trading]].
  
Unlike Arithmetic Rewards for Investments, the functions used forGeometric Rewards for Investments to determine the relationship between the Resources and Rewards of Investments are not linear. Thereby, these types of Investments are incompatible with both Arithmetic Rewards for Investments and Diminishing Returns. Using Geometric Rewards for Investments together with increasing the Risk/Reward factor gives the players more interesting choices, by increasing the uncertainties for getting the Reward. This can be used as a sort of a player perceived Balancing Effect where the players who are lagging behind in the progress have a chance to reach the leading players but with the drawback of increased risks. Of course, the leading players can also increase the lead in a similar fashion unless some other kind of Balancing Effect is introduced, for example, by not allowing the leading players to make these kinds of Investments.
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Whenever [[Geometric Progression]] and [[Risk/Reward]] both relate to a game event, the former affects the latter. This since either the reward becomes larger or smaller in relation to the risk, or, if the risk and the reward aspects both change by the same amount, the outcome will have larger or smaller effect on the gameplay overall. When applied on [[Investments]], [[Geometric Progression]] provides [[Stimulated Planning]] either to collect the necessary resources to achieve [[Combos]] or to avoid [[Diminishing Returns]]. When the [[Investments]] can provide [[Positive Feedback Loops]] this can require [[Timing]] and trying to succeed with this can easily cause [[Tension]].
  
Geometric Rewards for Investments instantiates Hovering Closure, especially when used together with Collecting. Compared to Arithmetic Rewards for Investments players are more likely to have strong commitments to subgoals with Geometric Rewards for Investments, and this may be used to increase Tension. Geometric Rewards for Investments also forces the players to plan the Timing of the Investments more carefully than with Arithmetic Rewards for Investments.
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Quite obviously, [[Geometric Progression]] is not applicable together with either [[Arithmetic Progression|Arithmetic]] or [[Discontinuous Progression]].
  
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== Relations ==
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=== Can Instantiate ===
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[[Negative Feedback Loops]],
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[[Positive Feedback Loops]],
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[[Stimulated Planning]]
  
=== Diegetic Aspects ===
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==== with [[Investments]] ====
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[[Stimulated Planning]]
  
=== Interface Aspects ===
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==== with [[Investments]] and [[Positive Feedback Loops]] ====
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[[Tension]],
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[[Timing]]
  
=== Narrative Aspects ===
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==== with [[Negative Feedback Loops]] ====
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[[Diminishing Returns]],
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[[Varied Gameplay]]
  
== Consequences ==
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==== with [[Extended Actions]] ====
[[Rewards]]
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[[Value of Effort]]
[[Penalties]]
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Geometric Rewards for Investments give the players an incentive to focus their Resources or attention to certain activities, for example Collecting more Resources of the same kind. This can also be used as an incentive for the players to do Trading with other players or other actions that have the effect of Transfer of Control events.
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The design choices with Geometric Rewards for Investment are mainly similar to Investments in general. The main differences, however, is the increased incitement to choose an artificial maximum amount that can be invested or to create an abrupt cutoff point where further Investments have Diminishing Returns. The main reason for the increased possibility for these design choices lie in the fact that Geometric Rewards for Investment gives Rewards that grow exponentially and create problems in Player Balance.
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Although Investments are often made by committing Resources, Extended Actions and Combos are actions that can merit Geometric Rewards for Investments.
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Quite obviously, [[Geometric Progression]] is not applicable together with either [[Arithmetic Progression|Arithmetic]] or [[Discontinuous Progression]].
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== Relations ==
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=== Can Instantiate ===
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==== with ... ====
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==== with [[Positive Feedback Loops]] ====
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[[Combos]],
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[[Higher-Level Closures as Gameplay Progresses]],
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[[Value of Effort]]
  
 
=== Can Modulate ===
 
=== Can Modulate ===
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[[Collecting]],
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[[Extended Actions]],
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[[Investments]],
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[[Penalties]],
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[[Risk/Reward]],
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[[Resources]],
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[[Rewards]],
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[[Scores]],
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[[Sets]],
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[[Trading]],
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[[Vulnerabilities]]
  
 
=== Can Be Instantiated By ===
 
=== Can Be Instantiated By ===
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[[Drawing Stacks]]
  
 
=== Can Be Modulated By ===
 
=== Can Be Modulated By ===
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-
  
 
=== Possible Closure Effects ===
 
=== Possible Closure Effects ===
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-
  
 
=== Potentially Conflicting With ===
 
=== Potentially Conflicting With ===
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<references>
 
<references>
 
<ref name="Bjork & Holopainen 2004">Björk, S. & Holopainen, J. (2004) ''Patterns in Game Design''. Charles River Media. ISBN1-58450-354-8.</ref>
 
<ref name="Bjork & Holopainen 2004">Björk, S. & Holopainen, J. (2004) ''Patterns in Game Design''. Charles River Media. ISBN1-58450-354-8.</ref>
 +
<ref name="wiki">Wikpipedia [http://en.wikipedia.org/wiki/Geometric_progression entry] for geometric progression.</ref>
 
</references>
 
</references>
  
 
== Acknowledgements ==
 
== Acknowledgements ==
 
-
 
-

Latest revision as of 10:31, 23 November 2015

Relations where each change in effort results in the effect being multiplied by a fixed value.

Geometric Progression occurs in games when the growth of effects are not linear but grows by depending on the current strengths of the effects and constant factor, called the common ratio. For factors larger than 1, Geometric Progression makes each new effort have a greater effect than the previous one and can thus discourages players to split resources into several smaller investments in favor of one large investment as the combined rewards of two lesser investments are always smaller than the reward of the larger investment using the same amount of resources. Factors less than 1 have the opposite effect and encourage players to spread resources.

For the purpose of this pattern, the common ratio is assumed to be larger than 0 (progression shifting from positive to negative values are rare in games) and differ from 1 (this would simply make extra effort have no effect). See wikipedia[1] for more information on geometric progression.

Examples

The point reward in Tetris for removing more than one row at once increases in a geometric fashion. The pot in the Poker variant Guts does not have to grow between rounds, but when it grows it grows by a factor 2 or more leading to a Geometric Progression in its size until it resets.

The rent one has to pay for landing on another players' property in Monopoly grows geometrically for the first three houses but tampers of for the fourth and for hotels. Exchanging collected cards for victory points in Bohnanza follows a Geometric Progression but in reducing the number of cards needs to get another point rather than providing more points. For example, 4 wax beans are required to get 1 points, 7 to get 2 points, 9 for 3 points, and 11 for 4 points.

The first levels for buying skills in the roleplaying system GURPS has a Geometric Progression that encourages players to diversify their skill sets.

Using the pattern

Geometric Progression can be included in games through setting progression schemes. These schemes are often applied to Resources through Investments and thereby potential to Rewards and Penalties, but can also directly modify Reward and Penalty schemes as well as how Extended Actions or Vulnerabilities function. As something affected by Reward and Penalty, Score systems can also be modified to make Geometric Progression present in the calculations when increasing or decreasing scores. Sets are another potential target for Geometric Progression, here the schemes apply to the number of game elements collected or equipped. An alternative is to consider changing the effectiveness of individual actions which has the same overall effect if the actions are aimed at the same target but requires prioritization of target order if many targets are chosen.

Some other patterns have progression quite close enough to geometric progression to be worthy mention. Drawing Stacks have the characteristic that the chance of drawing a specific Card increase for each extra Card drawn, and the increase can for all but the first extra Card be describes as a multiplier between 1 and 1.5. This can be used to create implicit Time Limits - Pandemic and Thunderstone uses this together with Stack Seeding to guarantee that certain cards appear semi-regularly.

Consequences

Geometric Progression typically provides Positive Feedback Loops or Negative Feedback Loops depending on if the common ratio is above 1 or below 1. With Positive Feedback Loops the pattern creates Combos and Higher-Level Closures as Gameplay Progresses, and thereby Value of Effort. Value of Effort can also be achieved in a similar way with Extended Actions, that of players creating an increasing value of effort as long as the action is maintained since the action becomes more and more powerful or rewarding the longer it is continued. In contrast, with Negative Feedback Loops players are instead encouraged to seek Varied Gameplay in order to avoid Diminishing Returns. Regardless of what type feedback loops created, this changes how players need to consider Collecting and Trading.

Whenever Geometric Progression and Risk/Reward both relate to a game event, the former affects the latter. This since either the reward becomes larger or smaller in relation to the risk, or, if the risk and the reward aspects both change by the same amount, the outcome will have larger or smaller effect on the gameplay overall. When applied on Investments, Geometric Progression provides Stimulated Planning either to collect the necessary resources to achieve Combos or to avoid Diminishing Returns. When the Investments can provide Positive Feedback Loops this can require Timing and trying to succeed with this can easily cause Tension.

Quite obviously, Geometric Progression is not applicable together with either Arithmetic or Discontinuous Progression.

Relations

Can Instantiate

Negative Feedback Loops, Positive Feedback Loops, Stimulated Planning

with Investments

Stimulated Planning

with Investments and Positive Feedback Loops

Tension, Timing

with Negative Feedback Loops

Diminishing Returns, Varied Gameplay

with Extended Actions

Value of Effort

with Positive Feedback Loops

Combos, Higher-Level Closures as Gameplay Progresses, Value of Effort

Can Modulate

Collecting, Extended Actions, Investments, Penalties, Risk/Reward, Resources, Rewards, Scores, Sets, Trading, Vulnerabilities

Can Be Instantiated By

Drawing Stacks

Can Be Modulated By

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Possible Closure Effects

-

Potentially Conflicting With

Arithmetic Progression, Discontinuous Progression

History

A renamed and updated version of the pattern Geometric Rewards for Investments that was part of the original collection in the book Patterns in Game Design[2].

References

  1. Wikpipedia entry for geometric progression.
  2. Björk, S. & Holopainen, J. (2004) Patterns in Game Design. Charles River Media. ISBN1-58450-354-8.

Acknowledgements

-