Game Theory is a tool used in several fields of science, including Political Science and Economics. For this article, a very simplified version will be used where the model consists of a rational agent (it always chooses to perform the action with the optimal expected outcome for itself from among all feasible actions) competing in a mathematically defined competition.
To put it more plainly, we are going to try and find the theoretical scoring maximums for each part of the game.
Using a game theory-based evaluation of the scoring system accomplishes three things:
- Make sure the students understand the scoring rules
- Positions optimized scoring strategies at the front of the discussion leading into system strategy brainstorming
- Provides the team with an objective reference that can be used as an unbiased evaluation of strategies in later phases of design
Check out our Ri3D Game Theory explainer video below:
To start this article, it is assumed that the reader is familiar with this season’s game and the corresponding game manual.
To begin, we will break the game into three sections for evaluation:
- Auto (20 Seconds)
- Teleoperation Mode (140 Seconds)
- End Game (T-minus 30 Seconds).
In each section, we ask: “What are all of the ways our robot can score points? Are any of these scoring methods mutually exclusive (for example, we can only score points for a single TOWER LEVEL)?”
For reference we will add a copy of “Figure 5-4 Areas, markings, and zones” from the game manual to clarify location terminology.
Figure 5-4
Autonomous Mode
In Autonomous Mode, teams stage their ROBOT on their ROBOT STARTING LINE.
The game manual (Table 6-4 REBUILT Point Values) details that a robot can score points in this mode by:
- FUEL – Scored in an active HUB (1 point)
- TOWER – Each ROBOT at LEVEL 1 (2 ROBOTS max)
A robot can be pre-loaded with up to eight (8) FUEL (per Section 6.4.3 (C) in the Game Manual).
The first decision we need to make is what we should do with our preloaded FUEL.
Treating the robot as a rational agent, it chooses to perform the actions with the optimal expected outcomes and scores eight (8) FUEL in the HUB (8 pts). That puts us at 3s of AUTO used with 8 pts scored.
Next, we assume it takes 2s to move from the ROBOT START LINE to the OUTPOST. If we assume an average speed of 10 ft/s for the robot, then evaluating the distance to the OUTPOST (~20ft), it takes approximately 2s to reach the OUTPOST. Next, we assume it takes 2s to collect 24 FUEL placed on the CHUTE. That puts us at 7s of AUTO used.
Choosing the shortest path to scoring, we move to line up to shoot into the HUB. We assume this will take 2s, then shoot our (24) FUEL (which we will assume takes 6s), scoring us 24 pts at 15 seconds of auto used.
Finally, we drive to the TOWER and perform a LEVEL 1 climb. This takes 5s, which finishes the total of our 20s in AUTO and earns us an additional 15 pts.
This MAX* Autonomous Mode achieves a score of 47 pts (count of 32 fuel and 1 level 2 climb).
Teleop Mode
In Teleoperation Mode, the robot is hanging on LEVEL 1 at the TOWER at the end of AUTO. The game manual (Table 6-4 REBUILT Point Values) details that a robot can score points in this mode by:
- FUEL scored in HUB (1 points)
Also, the game manual (Table 6-4 REBUILT Point Values) details that a robot (or robots) can also score Ranking Points (RP) by:
- ENERGIZED (1 RP)
- SUPERCHARGED (1 RP)
The teleoperated period this year is broken up into 6 phases, each with different HUBS active, as described in the game manual (Table 6-3: Hub Status during MATCH Timeframes). It begins with a 10-second TRANSITION SHIFT, where both HUBS are active, then goes into four 25-second shifts that alternate between the two alliances having active HUBS. The alliance that scored the most points in the Autonomous period will have its HUB inactive in SHIFT 1. Since the rational agent is maximizing scoring, we can assume that their HUB will be inactive in SHIFT 1.
Treating the robot as a rational agent, it chooses to perform the actions with the optimal expected outcomes. Although their HUB is active during the TRANSITION SHIFT, we currently do not have any FUEL loaded into our robot. Therefore, they will climb back down the TOWER (2s) and traverse to the center of the field (~27″ at 10ft/s is 3s) to start collecting FUEL as the time turns to SHIFT 1. We can intake 24 FUEL from the center of the field (8s), which serves the added benefit of removing these FUEL as an option for our opponents.
During SHIFT 1, our HUB is inactive, and the rational agent cannot score. Therefore, the optimal path is to play defense on the opposing alliance, either by actively disrupting their robots or removing FUEL from near them so they have a harder time gaining FUEL. With 5s remaining in SHIFT 1, we travel back to our ALLIANCE ZONE (~50″ at 10ft/s is 5s) and line up to start scoring as soon as SHIFT 2 begins.
As soon as our HUB becomes active, our rational agent shoots our 24 loaded FUEL (6s). Since we have not loaded from the depot yet, we move to the depot (1s) and intake all 24 FUEL there (8s). We then line up (2s) and shoot our loaded FUEL again (6s). These actions take us 23s for 48 pts. We finish SHIFT 2 into SHIFT 3 by traversing back to the middle of the field (~27″ at 10ft/s is 3s).
SHIFT 3 will have the exact same actions as SHIFT 1, intaking another 24 FUEL (7s), playing defense, and travelling back to our ALLIANCE ZONE (~50″ at 10ft/s is 5s) and line up to start scoring as soon as SHIFT 4 begins.
Again, as soon as our HUB becomes active in SHIFT 4, our rational agent shoots our 24 loaded FUEL (6s). All remaining FUEL is now in the middle of the field, so we traverse back to the middle of the field (~27″ at 10ft/s is 3s) and intake another 24 FUEL (8s). We traverse back to our side of the field (3s), line up (2s) and shoot our 24 FUEL once more (6s). These actions together give us an additional 48 pts while taking us 28s, bringing us past SHIFT 4 and into the END GAME.
During the TELEOP period we have scored 96 FUEL. Therefore, the MAX TELEOP Mode using this model achieves a score of 96 pts.
End Game
During the END GAME in the final 30 seconds, both HUBS are active, giving us time to finish shooting the FUEL we started in SHIFT 4.
The game manual (Table 6-4 REBUILT Point Values) details that a robot can score additional points in the end game phase by:
- LEVEL 1 on TOWER (10 points)
- LEVEL 2 on TOWER (20 points)
- LEVEL 3 on TOWER (30 points)
Once our rational agent has finished shooting, there are 27 seconds remaining. We have enough time to either perform another cycle of intaking and scoring 24 more FUEL or climbing the TOWER. Since our additional cycle would give us fewer points than climbing to LEVEL 3 of the TOWER, we turn and drive (1s) and line up (2s) at the TOWER, and hang at LEVEL 3 which we assume will take 12s. This takes a total of 15s (total time 128) and scores us 30 pts.
This MAX End Game achieves a score of 30 pts.
Final Score
- MAX AUTO Mode: 47 pts
- MAX TELEOP Mode: 96 pts
- MAX End Game: 40 pts
MAX TOTAL: 173 pts
RP Breakdown
- Fuel Scored: 128
- Tower Points: 45
This is enough to give us the ENERGIZED RP.
NOTE: While we assume we are playing “by ourselves” in an imagined scenario where both of the other robots are just KOP Chassis bots, we cannot safely assume in any given match that we will still gain at least 5pts each the other bots climbing the TOWER at any level so we don’t have enough for the TRAVERAL RP.
Summary
So what good has this exercise done? By walking through the match as a rational agent, the flow of a match can be better understood. By picking the optimal actions, necessary robot functions begin to take shape. By approaching the game systematically, rules and strategic advantages provided by those rules can be uncovered.
This exercise has also provided a baseline to evaluate other game strategies against and to help define robot functions from.
Check out our next article: Part 2 – Robot Requirements for more fun!
stay denk friends
If you want to learn more about this process, check out our presentation from the 2018 Purdue FIRST Forums on Robot Requirements!
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