Not receiving a ticket does not dismiss your role in the accident. Perhaps you turned without signaling, stopped abruptly, failed to yield, or were speeding too fast for conditions. These are examples of actions (or inactions) contributing to multi-vehicle crashes.
As a third-party claimant, you can request damages from the other driver's insurance company if your role in the crash is below 50%. However, comparative negligence law reduces the amount you can receive based on your role in the accident.
The auto insurance adjuster is tasked with conducting a detailed claim investigation to assess fault. This may involve driver statements, damage inspections, a crash report, and more. Perhaps you're in a situation where the other driver's insurer offers to pay 80% of your vehicle repair bill. This means the adjuster concluded you were 20% responsible for the crash.
Talk to the insurance adjuster if you disagree with the settlement offer. Or, turn the loss into your own company if you have collision coverage and damages exceed the deductible. Ultimately, only a court can determine how much you're owed.