Source-linked AI summary
Rock, Paper, Scissors, ... Dynamite - A Model of Disruption from New Technologies
Andrew J. Lohn
TL;DR
The paper asks how highly capable new technologies alter competition and uses Dynamite added to Rock-Paper-Scissors as a simplified model. It analyzes different game sizes, capability levels, and access patterns, finding that disruption can make prior moves unplayable while the new move’s value and usage remain limited. The model therefore highlights a gap between capability, strategic effect, and observed usage.
Problem
The paper addresses limited mechanisms for anticipating how versatile new technologies, including AI, affect competitions and create competitive advantage.
Method
The paper models Dynamite with varying capabilities in original and expanded Rock-Paper-Scissors games, considering equal, unequal, and one-sided access.
Results
The model shows that new moves can make prior options strategically unplayable, while capability has decreasing marginal value and high-capability moves are played rarely.
Takeaways & Limitations
Strategic disruption and value can arise without proportional use of the new move, and some apparently redundant prior options can remain or become more important.
Takeaways & Limitations
The analysis restricts expanded games to new moves that lose to a single opponent move while allowing variation in how many moves they defeat.
Abstract
from arXiv · showhide
We seek to understand the effect of adding disruptive highly-capable new technologies to competitions by assessing the addition of Dynamite to Rock-Paper-Scissors. We find that providing a versatile Dynamite move to only one player provides limited value (win probability increases from 50% to 55.5%) and is played rarely. That value decreases further if the game is expanded beyond just the original three moves. We also observe several mechanisms by which prior moves can become strategically unplayable, or obsolete. We hope that this model illustrates some non-intuitive aspects of developing new versatile technologies. We also hope that it illustrates some pitfalls for developers and integrators to avoid in order to create value rather than merely capability.
1 Introduction
The paper uses Dynamite as a simple model of highly capable new technologies that can disrupt competitions without proportionate competitive value. It examines how new options reshape playability, value, usage, and the relative importance of existing options.
- Motivation: The model treats Dynamite as an analogy for versatile technologies that can defeat multiple opposing options across AI, military, sports, and corporate settings.The paper studies the adjective “dynamite”—capability across varied circumstances—rather than the game or noun itself.
- Strategic disruption: A new move may make prior moves strategically unplayable without changing expected outcomes or adding competitive value.Such changes can alter strategies and competition dynamics even when the overall outcome remains unchanged.
- Capability and value: Increasing capability has decreasing marginal value and eventually saturates, while reducing the number of opposing moves that defeat the new move continually increases its value.The paper also considers unequal access, where limited Dynamite can be more valuable than enhanced Dynamite because the new move breaks game symmetry.
- Capability and value: The high-capability move is played rarely, yet its presence changes opponents’ strategies and can make less capable moves more frequently useful.Consequently, usage rates may not capture the technology’s competitive value, requiring alternative approaches to valuation.
- Strategic disruption: A new high-capability move can create obsolescence through domination or by producing optimal strategies that exclude otherwise non-dominated prior moves.The latter mechanism is prevalent in the Rock-Paper-Scissors structure.
- Strategic disruption: Some prior moves remain playable and can increase in usage dramatically even when the new move also defeats the only opposing move they target.This shows that apparent redundancy does not necessarily imply declining strategic importance.
2 Dynamite and Dominance
Adding Dynamite can alter strategies and make an original move obsolete without necessarily changing expected outcomes. With impervious Dynamite, outcomes become certain draws when both players have it, while weaker configurations can leave the game’s dynamics unchanged.
- If Dynamite defeats all three original moves and loses to none, one-sided access guarantees wins, while shared access produces Dynamite-Dynamite draws.
- If Dynamite loses to both Paper and Scissors, it is weakly dominated by Paper and should never be played.
- If Dynamite defeats Rock and Paper but loses to Scissors, it weakly dominates Paper, making Paper obsolete while preserving Rock-Scissors-Dynamite dynamics.
- Adding Dynamite can change strategies and make prior moves unplayable without changing the competition’s expected payoff.
- Subsequent analysis examines how asymmetric access and larger contests produce more substantial competitive changes.
3 Dynamite Denial
When only one competitor has Dynamite that defeats Rock and Paper, the new move creates a modest advantage through strategic pressure rather than frequent Dynamite use. The denied competitor uses Scissors more often, allowing the Dynamite-holder to exploit that shift with Rock.
- In this asymmetric configuration, Paper remains dominated for the Dynamite-holder but remains useful for the Dynamite-denied competitor.
- 55.6% is the Dynamite-holder’s equilibrium win probability, compared with 50% without asymmetric access.
- The Dynamite-denied competitor increases Scissors use, which the Dynamite-holder exploits by using Rock more often.
- Dynamite does not deliver an outsized share of the Dynamite-holder’s wins; its threat changes the opponent’s behavior instead.
4 Expanded Rock, Paper, Scissors
Expanding the game beyond three moves creates one-to-one and many-to-many competition structures with different patterns of wins, losses, and draws. Dynamite breaks the original symmetry, and its dominance and obsolescence effects depend on the expanded payoff structure.
- Expanded competitions can be one-to-one, where each move defeats and loses to one move, or many-to-many, as in Rock-Paper-Scissors-Lizard-Spock.
- The analysis restricts Dynamite to losing against one opponent move while varying how many opponent moves it defeats.
- Dynamite breaks symmetry by changing move utility according to whether moves defeat, draw against, or lose to it.
- In one-to-one contests, a Dynamite defeated by an opponent move can dominate only the one move also defeated by that opponent move.
- In many-to-many contests, Dynamite’s dominance depends on which opponent moves it defeats, and it may dominate nothing.
- The number of moves made unplayable or obsolete need not be small merely because Dynamite dominates few moves directly.
5 Dynamite Denial in an Expanded One-to-One Competition
The expanded one-to-one model shows that Dynamite’s value depends on the strategically playable moves, not simply the total arsenal or raw capability. Increasing capability can yield diminishing or zero additional value, while many apparently useful moves become strategically unplayable and Dynamite itself is often rarely used.
- 5.1 Defining Dynamite: Dynamite’s value depends on n, the initially available moves, k−1, the moves it does not defeat, and the strategically optimal move count m.The model replaces n with m because not all available moves belong in the equilibrium support.
- 5.2 Tactically Useful, Strategically Unplayable: For a given k, increasing m initially raises value v, but further increases eventually reduce it after a peak.The optimal Dynamite-holder strategy includes k−1 moves targeting moves Dynamite cannot defeat and j = m−k+1 additional non-Dynamite moves when available.
- 5.2 Tactically Useful, Strategically Unplayable: Moves outside the equilibrium support can remain tactically useful yet become strategically dominated because including them lowers the odds of victory.This creates a form of obsolescence without direct domination by Dynamite.
- 5.3 Dynamite Value: Increasing the number of moves Dynamite defeats raises its value only while n < m; beyond that point, additional defeated moves add no value.Reducing k, the number of moves Dynamite does not defeat, continues to increase its value.
- 5.3 Dynamite Value: When n > m and k >> 1, the value of Dynamite falls quadratically as the number of opponent moves it does not defeat increases.This approximation describes large contests with many available moves and many non-defeated moves.
- 5.4 Dynamite Usage Rates: Dynamite is often played rarely, sometimes nearly least often, even though its existence can increase its potential probability relative to the no-Dynamite baseline.Its strategic value may come from changing the opponent’s behavior rather than from frequent direct use, complicating accounting and pricing by usage.
6 Expanded Many-to-Many RPS
In the five-move Rock-Paper-Scissors-Lizard-Spock extension, adding Dynamite makes some moves strategically unplayable even though Dynamite does not dominate them. The Nash equilibrium excludes Spock and Lizard, while Dynamite is the least-used playable move.
- 6 Expanded Many-to-Many RPS: Dynamite is configured in the five-move game to defeat three moves, lose to one, and draw against itself and one other.The example extends the one-to-one analysis to a many-to-many payoff structure.
- 6 Expanded Many-to-Many RPS: Although all five moves are equally necessary without Dynamite, the Nash equilibrium after its addition excludes Spock and Lizard.Neither excluded move is dominated by Dynamite in the payoff matrix, but both become strategically unplayable.
- 6 Expanded Many-to-Many RPS: Dynamite becomes the least-used move among the moves that remain playable.Its low usage accompanies the strategic removal of two other moves.
7 Dynamite vs Lesser Dynamite
With unequal access to Dynamite, a less capable move can outperform a more capable one because it targets strategically critical options and compensates for its own weaknesses. Limited Dynamite can also make additional moves obsolete and reduce the game’s value.
- 7 Dynamite vs Lesser Dynamite: In the seven-move baseline, the Dynamite-extensive competitor defeats three moves against a fully Dynamite-denied opponent and gains value 1/35, or about 0.029.Figure 1 represents the extensive competitor’s wins and losses against each denied move.
- 7 Dynamite vs Lesser Dynamite: Giving the limited competitor Dynamite that defeats a1 and a6 makes a6 fall outside the support, which also removes b6 because b6 only defeats a6.The limited move changes both competitors’ strategically playable sets.
- 7 Dynamite vs Lesser Dynamite: The game’s value decreases from 1/35 to 1/45, or from 0.029 to 0.022, after limited Dynamite makes additional moves obsolete.The newly unplayable moves were not obsolete under the more capable Dynamite alone.
- 7 Dynamite vs Lesser Dynamite: The limited competitor can be better positioned than the extensive competitor when its Dynamite targets a critical move that helps defend against the stronger Dynamite.In the a1-and-a3 case, the extensive competitor’s value is negative 1/56, about −0.018, and the limited competitor expects to win.
- 7 Dynamite vs Lesser Dynamite: A less capable Dynamite can be more valuable than an enhanced Dynamite because targeting strategically important options can compensate for its smaller defeat set.The paper also notes that unplayable moves can differ between competitors even when limited Dynamite defeats a subset of the moves affected by extensive Dynamite.
8 Conclusions
The Dynamite model shows that highly capable additions can disrupt competition without adding value or changing expected outcomes, and that their value may be limited in larger games. It also identifies strategic unplayability, non-intuitive effects, and pricing implications for new technologies.
- New additions can disrupt competition without adding value or changing expected outcomes.
- The value of a disruptive addition can be lower than anticipated, especially in games with many possible moves.
- Because the addition's value typically does not come from frequent use, developers may need alternative approaches to assigning value and pricing new technology.
- Prior moves can become strategically unplayable through dominance or removal from support, illustrating mechanisms of technological obsolescence.
- Some moves remain playable despite redundancy and may increase in usage, sometimes dramatically, when new highly capable options are introduced.
- A less capable addition for a weaker competitor can make prior moves unplayable and may sometimes be more valuable than a more capable addition.
A Nash Equilibrium for Dynamite in Expanded One-to-One Rock-Paper-Scissors
The analysis derives a Nash equilibrium for expanded one-to-one Rock-Paper-Scissors with Dynamite by solving payoff-based probability conditions and optimizing which moves remain playable. It finds that, when some moves fall outside support, game value scales approximately with the inverse square of k.
- At Nash equilibrium, every move has the same expected value under the opponent's choice distribution.
- The payoff matrix separates moves Dynamite draws against from moves it defeats, yielding different equilibrium equations across the two regions.
- The equilibrium probabilities are determined by interface, normalization, and cyclic conditions applied to the region-specific solution.
- The Dynamite-holder retains non-Dynamite moves to maximize game value, with m determined by the number of Dynamite-defeated moves indexed by j.
- The optimal j is obtained by setting the first derivative to zero, while treating j as real to expose a simple square-root relationship for intuition.
- When sufficiently many moves exist for some to fall outside support, game value scales approximately with the inverse square of k as k increases.
B Limited Dynamite in a Matched-Moves One-to-One Competition
The matched-moves competition examines unequal Dynamite capabilities in a seven-move game. A more extensive Dynamite move yields a small quantitative advantage, while capability differences can remove different prior moves from equilibrium support.
- The seven-move matched-moves model gives each competitor equivalent moves that draw against themselves, defeat one move, and lose to another.
- Both competitors initially use Dynamite that defeats the first move, loses to the sixth, and draws against itself; one competitor then gains Dynamite that also defeats the second and third moves.
- 1/28 or 0.0357 is the expected winning value for the competitor with extensive Dynamite, with all moves played by both competitors.
- 1/33 or 0.03 is the extensive-Dynamite competitor's expected winning value when limited Dynamite defeats moves one and two.
- The capability-subset case removes one move from each competitor's support, but the unplayable moves differ between competitors.