Geometric Progression

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 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 beens 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 through setting progression schemes. These schemes are often applied to Investments and thereby to potential Rewards and Penalties, but can also directly modify Reward and Penalty schemes as well as how Extended Actions function. 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.

Randomness

Unlike Arithmetic Rewards for Investments, the functions used for Geometric 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.

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.

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.

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. 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.

When applied on Investments, Geometric Progression provides Stimulated Planning either to collect the necessary resources to achieve Combos or to avoid Diminishing Returns.

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 Negative Feedback Loops

Diminishing Returns, Varied Gameplay

with Positive Feedback Loops

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

Can Modulate

Collecting, Extended Actions, Investments, Penalties, Rewards, Trading

Can Be Instantiated By

Drawing Stacks

Can Be Modulated By

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Possible Closure Effects

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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.

Acknowledgements

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